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Guzmán–Neilan

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Abbreviated namesBR
Orders\(k=1\)
Reference elementstriangle, tetrahedron
DOFsOn each edge: (if \(k>1\)) point evaluations in tangential directions at midpoints
On each facet: point evaluations in normal directions at vertices,
normal integral moments with an order \(k-1\) Lagrange space, and (if \(k>1\)) point evaluations in normal directions at midpoints of edges
Number of DOFstriangle: \(11\)
tetrahedron: \(\begin{cases}19&k=1\\49&k=2\end{cases}\)
NotesThis element is a modification of the (/elements/bernardi-raugel.html)[Bernardi–Raugel] element with the facet bubbles modified to be divergence free.
CategoriesVector-valued elements, H(div) conforming elements

Implementations

Symfem"Guzman-Neilan"
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Examples

triangle
order 1
tetrahedron
order 1
tetrahedron
order 2
  • \(R\) is the reference triangle. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(\begin{cases} \left(\begin{array}{c}\displaystyle - x - 2 y + 1\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - 2 x - y + 1\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle - x - 2 y + 1\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle - 2 x - y + 1\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle x - y\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 2 x + y - 1\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle x - y\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 x + y - 1\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - x + y\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle x + 2 y - 1\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle - x + y\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle x + 2 y - 1\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 3 y\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 3 x\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - 3 x - 3 y + 3\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 3 y\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 3 x\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle - 3 x - 3 y + 3\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle \frac{y \left(- 6 x + 9 y - 4\right)}{2}\\\displaystyle \frac{y \left(3 y - 4\right)}{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle \frac{x \left(3 x - 4\right)}{2}\\\displaystyle \frac{x \left(9 x - 6 y - 4\right)}{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle \frac{3 x^{2}}{2} - 3 x y - x - \frac{3 y^{2}}{2} + 2 y - \frac{1}{2}\\\displaystyle - \frac{3 x^{2}}{2} - 3 x y + 2 x + \frac{3 y^{2}}{2} - y - \frac{1}{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle y \left(3 y - 4\right)\\\displaystyle 2 y\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - 3 x^{2} + 2 x + 6 y^{2} - 6 y\\\displaystyle 2 x \left(- 3 x + 3 y + 1\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 9 x^{2} + 24 x y - 14 x + 15 y^{2} - 20 y + 5\\\displaystyle - 6 x^{2} - 18 x y + 10 x - 12 y^{2} + 16 y - 4\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 2 y \left(- 3 x + 3 y - 1\right)\\\displaystyle - 6 x^{2} + 6 x + 3 y^{2} - 2 y\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle - 2 x\\\displaystyle x \left(4 - 3 x\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\left(\begin{array}{c}\displaystyle 12 x^{2} + 18 x y - 16 x + 6 y^{2} - 10 y + 4\\\displaystyle - 15 x^{2} - 24 x y + 20 x - 9 y^{2} + 14 y - 5\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\)
  • \(\mathcal{L}=\{l_0,...,l_{10}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0)\cdot\left(\begin{array}{c}-1\\-1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle \begin{cases} \tfrac{3 x y}{2} - x - \tfrac{9 y^{2}}{4} + \tfrac{5 y}{4}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{x \left(1 - 3 x\right)}{4}&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\- \tfrac{3 x^{2}}{4} + \tfrac{3 x y}{2} - \tfrac{3 x}{4} + \tfrac{3 y^{2}}{4} - \tfrac{5 y}{4} + \tfrac{1}{2}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\\\displaystyle \begin{cases} \tfrac{y \left(1 - 3 y\right)}{4}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{x \left(- 9 x + 6 y + 1\right)}{4}&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\tfrac{3 x^{2}}{4} + \tfrac{3 x y}{2} - \tfrac{x}{4} - \tfrac{3 y^{2}}{4} + \tfrac{5 y}{4} - \tfrac{1}{2}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\end{array}\right)\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1)\cdot\left(\begin{array}{c}-1\\-1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle \begin{cases} \tfrac{y \left(6 x - 9 y + 1\right)}{4}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{x \left(1 - 3 x\right)}{4}&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\- \tfrac{3 x^{2}}{4} + \tfrac{3 x y}{2} + \tfrac{5 x}{4} + \tfrac{3 y^{2}}{4} - \tfrac{y}{4} - \tfrac{1}{2}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\\\displaystyle \begin{cases} \tfrac{y \left(1 - 3 y\right)}{4}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- \tfrac{9 x^{2}}{4} + \tfrac{3 x y}{2} + \tfrac{5 x}{4} - y&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\tfrac{3 x^{2}}{4} + \tfrac{3 x y}{2} - \tfrac{5 x}{4} - \tfrac{3 y^{2}}{4} - \tfrac{3 y}{4} + \tfrac{1}{2}&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\end{array}\right)\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0)\cdot\left(\begin{array}{c}-1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle \begin{cases} x - \tfrac{3 y^{2}}{2} + \tfrac{5 y}{2} - 1&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{3 x^{2}}{2} - \tfrac{x}{2} - 3 y^{2} + 4 y - 1&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\- \tfrac{9 x^{2}}{2} - 12 x y + \tfrac{17 x}{2} - \tfrac{15 y^{2}}{2} + \tfrac{23 y}{2} - 4&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\3 x \left(x - y\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\3 x^{2} + 9 x y - 6 x + 6 y^{2} - 9 y + 3&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1)\cdot\left(\begin{array}{c}-1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle \begin{cases} \tfrac{y \left(1 - 3 y\right)}{2}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{3 x^{2}}{2} - \tfrac{3 x}{2} - 3 y^{2} + 2 y&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\- \tfrac{9 x^{2}}{2} - 12 x y + \tfrac{15 x}{2} - \tfrac{15 y^{2}}{2} + \tfrac{19 y}{2} - 3&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\3 x^{2} - 3 x y - x + y&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\3 x^{2} + 9 x y - 5 x + 6 y^{2} - 7 y + 2&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle \begin{cases} 3 y \left(x - y\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\0&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\- 6 x^{2} - 9 x y + 9 x - 3 y^{2} + 6 y - 3&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\\\displaystyle \begin{cases} 3 x^{2} - 4 x - \tfrac{3 y^{2}}{2} + \tfrac{y}{2} + 1&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{3 x^{2}}{2} - \tfrac{5 x}{2} - y + 1&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\tfrac{15 x^{2}}{2} + 12 x y - \tfrac{23 x}{2} + \tfrac{9 y^{2}}{2} - \tfrac{17 y}{2} + 4&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle \begin{cases} 3 x y - x - 3 y^{2} + y&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\0&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\- 6 x^{2} - 9 x y + 7 x - 3 y^{2} + 5 y - 2&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\\\displaystyle \begin{cases} 3 x^{2} - 2 x - \tfrac{3 y^{2}}{2} + \tfrac{3 y}{2}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{x \left(3 x - 1\right)}{2}&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\tfrac{15 x^{2}}{2} + 12 x y - \tfrac{19 x}{2} + \tfrac{9 y^{2}}{2} - \tfrac{15 y}{2} + 3&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{0}\)
where \(e_{0}\) is the 0th edge;
and \(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0.

\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle \begin{cases} \tfrac{y \left(- 6 x + 9 y - 1\right)}{2}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{x \left(3 x - 1\right)}{2}&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\\tfrac{3 x^{2}}{2} - 3 x y - \tfrac{5 x}{2} - \tfrac{3 y^{2}}{2} + \tfrac{y}{2} + 1&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\\\displaystyle \begin{cases} \tfrac{y \left(3 y - 1\right)}{2}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\\tfrac{x \left(9 x - 6 y - 1\right)}{2}&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\- \tfrac{3 x^{2}}{2} - 3 x y + \tfrac{x}{2} + \tfrac{3 y^{2}}{2} - \tfrac{5 y}{2} + 1&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\end{array}\right)\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{1}\)
where \(e_{1}\) is the 1st edge;
and \(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1.

\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle \begin{cases} y \left(3 y - 1\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- 3 x^{2} + 5 x + 6 y^{2} - 6 y&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\9 x^{2} + 24 x y - 17 x + 15 y^{2} - 23 y + 8&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\6 x \left(- x + y\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\- 6 x^{2} - 18 x y + 12 x - 12 y^{2} + 18 y - 6&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{2}\)
where \(e_{2}\) is the 2nd edge;
and \(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2.

\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle \begin{cases} 6 y \left(- x + y\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\0&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\12 x^{2} + 18 x y - 18 x + 6 y^{2} - 12 y + 6&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\\\displaystyle \begin{cases} - 6 x^{2} + 6 x + 3 y^{2} - 5 y&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\x \left(1 - 3 x\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\- 15 x^{2} - 24 x y + 23 x - 9 y^{2} + 17 y - 8&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{9}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{3},\tfrac{1}{3})\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle \begin{cases} 3 y&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\3 x&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\- 3 x - 3 y + 3&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\0&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\0&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{10}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{3},\tfrac{1}{3})\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\0&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\0&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\\\displaystyle \begin{cases} 3 y&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\3 x&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (1/3, 1/3)))\\- 3 x - 3 y + 3&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\end{cases}\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
  • \(R\) is the reference tetrahedron. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(\begin{cases} \left(\begin{array}{c}\displaystyle - x - y - 2 z + 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - x - 2 y - z + 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - 2 x - y - z + 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle - x - y - 2 z + 1\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle - x - 2 y - z + 1\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle - 2 x - y - z + 1\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - x - y - 2 z + 1\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - x - 2 y - z + 1\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - 2 x - y - z + 1\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle x - z\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle x - y\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 2 x + y + z - 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle x - z\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle x - y\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 x + y + z - 1\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x - z\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x - y\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 2 x + y + z - 1\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle y - z\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - x + y\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle x + 2 y + z - 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle y - z\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle - x + y\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle x + 2 y + z - 1\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y - z\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - x + y\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x + 2 y + z - 1\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - y + z\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - x + z\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle x + y + 2 z - 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle - y + z\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle - x + z\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle x + y + 2 z - 1\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - y + z\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - x + z\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x + y + 2 z - 1\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 4 z\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 4 y\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 4 x\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - 4 x - 4 y - 4 z + 4\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 z\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 y\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 x\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle - 4 x - 4 y - 4 z + 4\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 z\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 y\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 x\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - 4 x - 4 y - 4 z + 4\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle \frac{z \left(- 183750 x^{2} - 143936 x y - 222439 x z + 174750 x + 221968 y z + 428657 z^{2} - 208875 z + 2250\right)}{375}\\\displaystyle \frac{z \left(- 163564 x y + 231782 x z - 183750 y^{2} - 202811 y z + 174750 y + 418843 z^{2} - 208875 z + 2250\right)}{375}\\\displaystyle \frac{2 z \left(132766 x z + 127859 y z + 70875 z^{2} - 87375 z + 1125\right)}{375}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \frac{y \left(- 183750 x^{2} - 109061 x y - 257314 x z + 174750 x + 371968 y^{2} + 278657 y z - 208875 y + 2250\right)}{375}\\\displaystyle \frac{y \left(68218 x y + 1125 y^{2} + 31157 y z - 34125 y + 2250\right)}{375}\\\displaystyle \frac{2 y \left(17234 x y + 115532 x z + 97141 y^{2} + 52843 y z - 34125 y + 48750 z^{2} - 53250 z + 1125\right)}{375}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \frac{x \left(1125 x^{2} + 78032 x y + 21343 x z - 34125 x + 2250\right)}{375}\\\displaystyle \frac{x \left(381782 x^{2} - 128689 x y + 268843 x z - 208875 x - 183750 y^{2} - 237686 y z + 174750 y + 2250\right)}{375}\\\displaystyle \frac{2 x \left(92234 x^{2} + 22141 x y + 62657 x z - 34125 x + 105718 y z + 48750 z^{2} - 53250 z + 1125\right)}{375}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 263 x^{3} + \frac{91468 x^{2} y}{125} + \frac{148157 x^{2} z}{125} - 699 x^{2} + \frac{277747 x y^{2}}{375} + 2166 x y z - \frac{515122 x y}{375} + \frac{504503 x z^{2}}{375} - \frac{741878 x z}{375} + 627 x + \frac{101968 y^{3}}{375} + \frac{377593 y^{2} z}{375} - \frac{277811 y^{2}}{375} + \frac{434282 y z^{2}}{375} - 1824 y z + \frac{247468 y}{375} + \frac{158657 z^{3}}{375} - \frac{391189 z^{2}}{375} + \frac{304157 z}{375} - 191\\\displaystyle \frac{111782 x^{3}}{375} + \frac{317003 x^{2} y}{375} + \frac{387407 x^{2} z}{375} - \frac{297439 x^{2}}{375} + \frac{101282 x y^{2}}{125} + 2166 x y z - \frac{554378 x y}{375} + \frac{424468 x z^{2}}{375} - 1824 x z + \frac{257282 x}{375} + 263 y^{3} + \frac{138343 y^{2} z}{125} - 699 y^{2} + \frac{465247 y z^{2}}{375} - \frac{702622 y z}{375} + 627 y + \frac{148843 z^{3}}{375} - \frac{371561 z^{2}}{375} + \frac{294343 z}{375} - 191\\\displaystyle - \frac{85532 x^{3}}{375} - \frac{231782 x^{2} y}{375} - \frac{612878 x^{2} z}{375} + \frac{237814 x^{2}}{375} - \frac{221968 x y^{2}}{375} - 3084 x y z + 1176 x y - \frac{283532 x z^{2}}{125} + \frac{1078628 x z}{375} - \frac{221282 x}{375} - \frac{75718 y^{3}}{375} - \frac{573622 y^{2} z}{375} + \frac{218186 y^{2}}{375} - \frac{273718 y z^{2}}{125} + \frac{1039372 y z}{375} - \frac{211468 y}{375} - 862 z^{3} + 1926 z^{2} - 1248 z + 184\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle \frac{z \left(- 20250 x^{2} - 59434 x y - 39791 x z + 22050 x - 9000 y^{2} + 16217 y z + 9000 y + 50158 z^{2} - 13725 z - 450\right)}{75}\\\displaystyle \frac{z \left(68434 x y - 7217 x z + 38250 y^{2} + 37091 y z - 36450 y - 45658 z^{2} + 15525 z - 450\right)}{75}\\\displaystyle \frac{z \left(- 13967 x z - 8533 y z + 900 z^{2} + 7200 z - 450\right)}{75}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \frac{y \left(- 20250 x^{2} - 133909 x y + 34684 x z + 22050 x + 97217 y^{2} - 30842 y z - 13725 y - 9000 z^{2} + 9000 z - 450\right)}{75}\\\displaystyle \frac{y \left(61217 x y + 29025 y^{2} + 658 y z - 20925 y - 450\right)}{75}\\\displaystyle \frac{y \left(67967 x y - 81934 x z - 36467 y^{2} + 46834 y z - 12600 y - 18000 z^{2} + 19800 z - 450\right)}{75}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - 303 x^{3} - \frac{34217 x^{2} y}{75} + \frac{12842 x^{2} z}{75} + 111 x^{2} - 6 x - 120 y^{2} z - 120 y z^{2} + 120 y z\\\displaystyle \frac{x \left(- 43217 x^{2} + 95209 x y - 9658 x z + 15525 x + 38250 y^{2} + 10316 y z - 36450 y - 450\right)}{75}\\\displaystyle \frac{x \left(31967 x^{2} - 467 x y - 27034 x z - 12600 x - 8066 y z - 18000 z^{2} + 19800 z - 450\right)}{75}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - 1923 x^{3} - \frac{107683 x^{2} y}{25} - \frac{154742 x^{2} z}{25} + 4719 x^{2} - \frac{243607 x y^{2}}{75} - 9246 x y z + \frac{520582 x y}{75} - \frac{431843 x z^{2}}{75} + \frac{708818 x z}{75} - 3687 x - \frac{64783 y^{3}}{75} - \frac{259408 y^{2} z}{75} + \frac{196841 y^{2}}{75} - \frac{306467 y z^{2}}{75} + 6744 y z - \frac{198883 y}{75} - \frac{111842 z^{3}}{75} + \frac{290959 z^{2}}{75} - \frac{245942 z}{75} + 891\\\displaystyle \frac{28783 x^{3}}{75} + \frac{51907 x^{2} y}{75} + \frac{83908 x^{2} z}{75} - \frac{78041 x^{2}}{75} + \frac{15883 x y^{2}}{25} + 2094 x y z - \frac{119182 x y}{75} + \frac{81467 x z^{2}}{75} - 2016 x z + \frac{70183 x}{75} + 327 y^{3} + \frac{34442 y^{2} z}{25} - 951 y^{2} + \frac{105143 y z^{2}}{75} - \frac{172418 y z}{75} + 903 y + \frac{26342 z^{3}}{75} - \frac{73159 z^{2}}{75} + \frac{67742 z}{75} - 279\\\displaystyle \frac{103967 x^{3}}{75} + \frac{243467 x^{2} y}{75} + \frac{380768 x^{2} z}{75} - \frac{256534 x^{2}}{75} + \frac{175033 x y^{2}}{75} + 7344 x y z - 5016 x y + \frac{128567 x z^{2}}{25} - \frac{588668 x z}{75} + \frac{201617 x}{75} + \frac{35533 y^{3}}{75} + \frac{170032 y^{2} z}{75} - \frac{119666 y^{2}}{75} + \frac{81133 y z^{2}}{25} - \frac{377932 y z}{75} + \frac{133183 y}{75} + 1452 z^{3} - 3576 z^{2} + 2778 z - 654\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 6 z \left(40 x^{2} + 72 x y + 6 x z - 44 x - 96 y z + 26 z^{2} + 28 z + 1\right)\\\displaystyle \frac{2 z \left(22500 x^{2} - 24262 x y + 45881 x z - 22500 x - 10762 y^{2} + 118857 y z + 6262 y - 119738 z^{2} + 3619 z + 1125\right)}{375}\\\displaystyle \frac{2 z \left(- 32869 x z - 29738 y z - 41869 z^{2} + 21619 z + 1125\right)}{375}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 6 y \left(40 x^{2} - 50 x y + 128 x z - 44 x - 16 y^{2} - 54 y z + 28 y + 1\right)\\\displaystyle 120 x^{2} z - \frac{1762 x y^{2}}{375} + 120 x z^{2} - 120 x z + \frac{52238 y^{3}}{375} - \frac{75524 y^{2} z}{375} + \frac{19762 y^{2}}{375} + 6 y\\\displaystyle \frac{2 y \left(55369 x y - 88238 x z - 15262 y^{2} - 22107 y z - 8119 y - 34238 z^{2} + 29738 z + 1125\right)}{375}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 6 x \left(- 2 x^{2} - 24 x y + 74 x z - 16 x + 1\right)\\\displaystyle \frac{2 x \left(- 156619 x^{2} + 192619 x y + 82762 x z + 3619 x - 10762 y^{2} - 98024 y z + 6262 y + 22500 z^{2} - 22500 z + 1125\right)}{375}\\\displaystyle \frac{2 x \left(145369 x^{2} - 105262 x y - 185869 x z - 8119 x + 75524 y z - 34238 z^{2} + 29738 z + 1125\right)}{375}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - 1452 x^{3} - 4152 x^{2} y - 4236 x^{2} z + 3576 x^{2} - 3756 x y^{2} - 7344 x y z + 6528 x y - 3588 x z^{2} + 6360 x z - 2778 x - 1056 y^{3} - 2916 y^{2} z + 2760 y^{2} - 2664 y z^{2} + 5016 y z - 2358 y - 804 z^{3} + 2256 z^{2} - 2106 z + 654\\\displaystyle \frac{496762 x^{3}}{375} + \frac{394162 x^{2} y}{75} + \frac{1654048 x^{2} z}{375} - \frac{463762 x^{2}}{125} + \frac{2379334 x y^{2}}{375} + \frac{4326668 x y z}{375} - \frac{732524 x y}{75} + \frac{345562 x z^{2}}{75} - \frac{3020096 x z}{375} + \frac{430012 x}{125} + \frac{301762 y^{3}}{125} + \frac{520124 y^{2} z}{75} - \frac{2199334 y^{2}}{375} + \frac{755286 y z^{2}}{125} - \frac{3957668 y z}{375} + \frac{337912 y}{75} + \frac{570524 z^{3}}{375} - \frac{307762 z^{2}}{75} + \frac{1363798 z}{375} - \frac{395512}{375}\\\displaystyle - \frac{69262 x^{3}}{375} - \frac{529048 x^{2} y}{375} - \frac{67462 x^{2} z}{75} + \frac{100762 x^{2}}{125} - \frac{170062 x y^{2}}{75} - \frac{1644668 x y z}{375} + \frac{1247096 x y}{375} - \frac{574834 x z^{2}}{375} + \frac{196124 x z}{75} - \frac{133012 x}{125} - \frac{390524 y^{3}}{375} - \frac{435786 y^{2} z}{125} + \frac{188962 y^{2}}{75} - \frac{244724 y z^{2}}{75} + \frac{1950668 y z}{375} - \frac{720298 y}{375} - \frac{102262 z^{3}}{125} + \frac{786334 z^{2}}{375} - \frac{129112 z}{75} + \frac{166012}{375}\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle \frac{z \left(343124 x^{2} + 421874 x y + 485436 x z - 334124 x - 233437 y z - 258749 z^{2} + 153562 z - 2250\right)}{375}\\\displaystyle 3 z \left(- 230 x y + 95 x z - 80 y^{2} + 74 y z + 88 y + 45 z^{2} - 56 z - 2\right)\\\displaystyle - 120 x^{2} y - 120 x y^{2} + 120 x y - \frac{213749 x z^{2}}{375} - \frac{120937 y z^{2}}{375} - \frac{189562 z^{3}}{375} + \frac{117562 z^{2}}{375} - 6 z\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \frac{y \left(343124 x^{2} + 479812 x y + 427498 x z - 334124 x - 413437 y^{2} - 78749 y z + 153562 y - 2250\right)}{375}\\\displaystyle 3 y \left(- 135 x y + 4 y^{2} + 35 y z + 32 y - 2\right)\\\displaystyle \frac{y \left(- 45000 x^{2} + 123749 x y - 382498 x z + 45000 x + 435937 y^{2} - 493312 y z - 144562 y - 253124 z^{2} + 262124 z - 2250\right)}{375}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \frac{x \left(221062 x^{2} + 188437 x y + 348749 x z - 180562 x - 2250\right)}{375}\\\displaystyle 3 x \left(- 65 x^{2} + 14 x y + 205 x z - 56 x - 80 y^{2} - 170 y z + 88 y - 2\right)\\\displaystyle \frac{x \left(528749 x^{2} + 30937 x y - 678936 x z - 144562 x - 45000 y^{2} - 196874 y z + 45000 y - 253124 z^{2} + 262124 z - 2250\right)}{375}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - \frac{35062 x^{3}}{125} - \frac{342559 x^{2} y}{375} - \frac{38699 x^{2} z}{75} + \frac{174934 x^{2}}{375} - \frac{58162 x y^{2}}{75} - \frac{277868 x y z}{375} + \frac{66824 x y}{75} + \frac{4314 x z^{2}}{125} + \frac{30368 x z}{375} - \frac{8212 x}{75} - \frac{53437 y^{3}}{375} - \frac{5623 y^{2} z}{375} + \frac{26812 y^{2}}{125} + \frac{29813 y z^{2}}{75} - \frac{148504 y z}{375} + \frac{563 y}{125} + \frac{101251 z^{3}}{375} - \frac{45788 z^{2}}{75} + \frac{156377 z}{375} - \frac{28688}{375}\\\displaystyle 765 x^{3} + 3432 x^{2} y + 2625 x^{2} z - 2178 x^{2} + 4119 x y^{2} + 7344 x y z - 6204 x y + 2955 x z^{2} - 5016 x z + 2067 x + 1452 y^{3} + 4269 y^{2} z - 3576 y^{2} + 3912 y z^{2} - 6684 y z + 2778 y + 1095 z^{3} - 2838 z^{2} + 2397 z - 654\\\displaystyle - \frac{281251 x^{3}}{375} - \frac{205313 x^{2} y}{75} - \frac{323814 x^{2} z}{125} + \frac{164588 x^{2}}{75} - \frac{1119377 x y^{2}}{375} - \frac{2404132 x y z}{375} + \frac{1921504 x y}{375} - \frac{236701 x z^{2}}{75} + \frac{1976632 x z}{375} - \frac{799877 x}{375} - \frac{374063 y^{3}}{375} - \frac{268538 y^{2} z}{75} + \frac{336188 y^{2}}{125} - \frac{1461941 y z^{2}}{375} + \frac{469576 y z}{75} - \frac{297563 y}{125} - \frac{164438 z^{3}}{125} + \frac{1238066 z^{2}}{375} - \frac{200588 z}{75} + \frac{258188}{375}\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)
  • \(\mathcal{L}=\{l_0,...,l_{18}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,0)\cdot\left(\begin{array}{c}1\\1\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle \begin{cases} \tfrac{245 x^{2} z}{3} + \tfrac{71968 x y z}{1125} + \tfrac{222439 x z^{2}}{2250} - \tfrac{233 x z}{3} + x - \tfrac{110984 y z^{2}}{1125} - \tfrac{428657 z^{3}}{2250} + \tfrac{557 z^{2}}{6} - 2 z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{245 x^{2} y}{3} + \tfrac{109061 x y^{2}}{2250} + \tfrac{128657 x y z}{1125} - \tfrac{233 x y}{3} + x - \tfrac{185984 y^{3}}{1125} - \tfrac{278657 y^{2} z}{2250} + \tfrac{557 y^{2}}{6} - 2 y&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(- 1125 x^{2} - 78032 x y - 21343 x z + 34125 x - 2250\right)}{2250}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{263 x^{3}}{6} - \tfrac{45734 x^{2} y}{375} - \tfrac{148157 x^{2} z}{750} + \tfrac{233 x^{2}}{2} - \tfrac{277747 x y^{2}}{2250} - 361 x y z + \tfrac{257561 x y}{1125} - \tfrac{504503 x z^{2}}{2250} + \tfrac{370939 x z}{1125} - \tfrac{205 x}{2} - \tfrac{50984 y^{3}}{1125} - \tfrac{377593 y^{2} z}{2250} + \tfrac{277811 y^{2}}{2250} - \tfrac{217141 y z^{2}}{1125} + 304 y z - \tfrac{122609 y}{1125} - \tfrac{158657 z^{3}}{2250} + \tfrac{391189 z^{2}}{2250} - \tfrac{301907 z}{2250} + \tfrac{185}{6}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(163564 x y - 231782 x z + 183750 y^{2} + 202811 y z - 174750 y - 418843 z^{2} + 208875 z - 2250\right)}{2250}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 68218 x y - 1125 y^{2} - 31157 y z + 34125 y - 2250\right)}{2250}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(- 381782 x^{2} + 128689 x y - 268843 x z + 208875 x + 183750 y^{2} + 237686 y z - 174750 y - 2250\right)}{2250}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{55891 x^{3}}{1125} - \tfrac{317003 x^{2} y}{2250} - \tfrac{387407 x^{2} z}{2250} + \tfrac{297439 x^{2}}{2250} - \tfrac{50641 x y^{2}}{375} - 361 x y z + \tfrac{277189 x y}{1125} - \tfrac{212234 x z^{2}}{1125} + 304 x z - \tfrac{128641 x}{1125} - \tfrac{263 y^{3}}{6} - \tfrac{138343 y^{2} z}{750} + \tfrac{233 y^{2}}{2} - \tfrac{465247 y z^{2}}{2250} + \tfrac{351311 y z}{1125} - \tfrac{209 y}{2} - \tfrac{148843 z^{3}}{2250} + \tfrac{371561 z^{2}}{2250} - \tfrac{294343 z}{2250} + \tfrac{191}{6}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(- 132766 x z - 127859 y z - 70875 z^{2} + 87375 z - 1125\right)}{1125}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 17234 x y - 115532 x z - 97141 y^{2} - 52843 y z + 34125 y - 48750 z^{2} + 53250 z - 1125\right)}{1125}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(- 92234 x^{2} - 22141 x y - 62657 x z + 34125 x - 105718 y z - 48750 z^{2} + 53250 z - 1125\right)}{1125}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{42766 x^{3}}{1125} + \tfrac{115891 x^{2} y}{1125} + \tfrac{306439 x^{2} z}{1125} - \tfrac{118907 x^{2}}{1125} + \tfrac{110984 x y^{2}}{1125} + 514 x y z - 196 x y + \tfrac{141766 x z^{2}}{375} - \tfrac{539314 x z}{1125} + \tfrac{110641 x}{1125} + \tfrac{37859 y^{3}}{1125} + \tfrac{286811 y^{2} z}{1125} - \tfrac{109093 y^{2}}{1125} + \tfrac{136859 y z^{2}}{375} - \tfrac{519686 y z}{1125} + \tfrac{105734 y}{1125} + \tfrac{431 z^{3}}{3} - 321 z^{2} + 208 z - \tfrac{92}{3}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,0)\cdot\left(\begin{array}{c}1\\1\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle \begin{cases} \tfrac{z \left(183750 x^{2} + 143936 x y + 222439 x z - 174750 x - 221968 y z - 428657 z^{2} + 208875 z - 2250\right)}{2250}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(183750 x^{2} + 109061 x y + 257314 x z - 174750 x - 371968 y^{2} - 278657 y z + 208875 y - 2250\right)}{2250}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(- 1125 x^{2} - 78032 x y - 21343 x z + 34125 x - 2250\right)}{2250}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{263 x^{3}}{6} - \tfrac{45734 x^{2} y}{375} - \tfrac{148157 x^{2} z}{750} + \tfrac{233 x^{2}}{2} - \tfrac{277747 x y^{2}}{2250} - 361 x y z + \tfrac{257561 x y}{1125} - \tfrac{504503 x z^{2}}{2250} + \tfrac{370939 x z}{1125} - \tfrac{209 x}{2} - \tfrac{50984 y^{3}}{1125} - \tfrac{377593 y^{2} z}{2250} + \tfrac{277811 y^{2}}{2250} - \tfrac{217141 y z^{2}}{1125} + 304 y z - \tfrac{123734 y}{1125} - \tfrac{158657 z^{3}}{2250} + \tfrac{391189 z^{2}}{2250} - \tfrac{304157 z}{2250} + \tfrac{191}{6}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{81782 x y z}{1125} - \tfrac{115891 x z^{2}}{1125} + \tfrac{245 y^{2} z}{3} + \tfrac{202811 y z^{2}}{2250} - \tfrac{233 y z}{3} + y - \tfrac{418843 z^{3}}{2250} + \tfrac{557 z^{2}}{6} - 2 z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 68218 x y - 1125 y^{2} - 31157 y z + 34125 y - 2250\right)}{2250}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{190891 x^{3}}{1125} + \tfrac{128689 x^{2} y}{2250} - \tfrac{268843 x^{2} z}{2250} + \tfrac{557 x^{2}}{6} + \tfrac{245 x y^{2}}{3} + \tfrac{118843 x y z}{1125} - \tfrac{233 x y}{3} - 2 x + y&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{55891 x^{3}}{1125} - \tfrac{317003 x^{2} y}{2250} - \tfrac{387407 x^{2} z}{2250} + \tfrac{297439 x^{2}}{2250} - \tfrac{50641 x y^{2}}{375} - 361 x y z + \tfrac{277189 x y}{1125} - \tfrac{212234 x z^{2}}{1125} + 304 x z - \tfrac{127516 x}{1125} - \tfrac{263 y^{3}}{6} - \tfrac{138343 y^{2} z}{750} + \tfrac{233 y^{2}}{2} - \tfrac{465247 y z^{2}}{2250} + \tfrac{351311 y z}{1125} - \tfrac{205 y}{2} - \tfrac{148843 z^{3}}{2250} + \tfrac{371561 z^{2}}{2250} - \tfrac{292093 z}{2250} + \tfrac{185}{6}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(- 132766 x z - 127859 y z - 70875 z^{2} + 87375 z - 1125\right)}{1125}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 17234 x y - 115532 x z - 97141 y^{2} - 52843 y z + 34125 y - 48750 z^{2} + 53250 z - 1125\right)}{1125}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(- 92234 x^{2} - 22141 x y - 62657 x z + 34125 x - 105718 y z - 48750 z^{2} + 53250 z - 1125\right)}{1125}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{42766 x^{3}}{1125} + \tfrac{115891 x^{2} y}{1125} + \tfrac{306439 x^{2} z}{1125} - \tfrac{118907 x^{2}}{1125} + \tfrac{110984 x y^{2}}{1125} + 514 x y z - 196 x y + \tfrac{141766 x z^{2}}{375} - \tfrac{539314 x z}{1125} + \tfrac{110641 x}{1125} + \tfrac{37859 y^{3}}{1125} + \tfrac{286811 y^{2} z}{1125} - \tfrac{109093 y^{2}}{1125} + \tfrac{136859 y z^{2}}{375} - \tfrac{519686 y z}{1125} + \tfrac{105734 y}{1125} + \tfrac{431 z^{3}}{3} - 321 z^{2} + 208 z - \tfrac{92}{3}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,1)\cdot\left(\begin{array}{c}1\\1\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle \begin{cases} \tfrac{z \left(183750 x^{2} + 143936 x y + 222439 x z - 174750 x - 221968 y z - 428657 z^{2} + 208875 z - 2250\right)}{2250}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(183750 x^{2} + 109061 x y + 257314 x z - 174750 x - 371968 y^{2} - 278657 y z + 208875 y - 2250\right)}{2250}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(- 1125 x^{2} - 78032 x y - 21343 x z + 34125 x - 2250\right)}{2250}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{263 x^{3}}{6} - \tfrac{45734 x^{2} y}{375} - \tfrac{148157 x^{2} z}{750} + \tfrac{233 x^{2}}{2} - \tfrac{277747 x y^{2}}{2250} - 361 x y z + \tfrac{257561 x y}{1125} - \tfrac{504503 x z^{2}}{2250} + \tfrac{370939 x z}{1125} - \tfrac{209 x}{2} - \tfrac{50984 y^{3}}{1125} - \tfrac{377593 y^{2} z}{2250} + \tfrac{277811 y^{2}}{2250} - \tfrac{217141 y z^{2}}{1125} + 304 y z - \tfrac{123734 y}{1125} - \tfrac{158657 z^{3}}{2250} + \tfrac{391189 z^{2}}{2250} - \tfrac{304157 z}{2250} + \tfrac{191}{6}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(163564 x y - 231782 x z + 183750 y^{2} + 202811 y z - 174750 y - 418843 z^{2} + 208875 z - 2250\right)}{2250}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 68218 x y - 1125 y^{2} - 31157 y z + 34125 y - 2250\right)}{2250}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(- 381782 x^{2} + 128689 x y - 268843 x z + 208875 x + 183750 y^{2} + 237686 y z - 174750 y - 2250\right)}{2250}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{55891 x^{3}}{1125} - \tfrac{317003 x^{2} y}{2250} - \tfrac{387407 x^{2} z}{2250} + \tfrac{297439 x^{2}}{2250} - \tfrac{50641 x y^{2}}{375} - 361 x y z + \tfrac{277189 x y}{1125} - \tfrac{212234 x z^{2}}{1125} + 304 x z - \tfrac{128641 x}{1125} - \tfrac{263 y^{3}}{6} - \tfrac{138343 y^{2} z}{750} + \tfrac{233 y^{2}}{2} - \tfrac{465247 y z^{2}}{2250} + \tfrac{351311 y z}{1125} - \tfrac{209 y}{2} - \tfrac{148843 z^{3}}{2250} + \tfrac{371561 z^{2}}{2250} - \tfrac{294343 z}{2250} + \tfrac{191}{6}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(- 132766 x z - 127859 y z - 70875 z^{2} + 87375 z - 1125\right)}{1125}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\- \tfrac{17234 x y^{2}}{1125} - \tfrac{115532 x y z}{1125} - \tfrac{97141 y^{3}}{1125} - \tfrac{52843 y^{2} z}{1125} + \tfrac{91 y^{2}}{3} - \tfrac{130 y z^{2}}{3} + \tfrac{142 y z}{3} - 2 y + z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{92234 x^{3}}{1125} - \tfrac{22141 x^{2} y}{1125} - \tfrac{62657 x^{2} z}{1125} + \tfrac{91 x^{2}}{3} - \tfrac{105718 x y z}{1125} - \tfrac{130 x z^{2}}{3} + \tfrac{142 x z}{3} - 2 x + z&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{42766 x^{3}}{1125} + \tfrac{115891 x^{2} y}{1125} + \tfrac{306439 x^{2} z}{1125} - \tfrac{118907 x^{2}}{1125} + \tfrac{110984 x y^{2}}{1125} + 514 x y z - 196 x y + \tfrac{141766 x z^{2}}{375} - \tfrac{539314 x z}{1125} + \tfrac{111766 x}{1125} + \tfrac{37859 y^{3}}{1125} + \tfrac{286811 y^{2} z}{1125} - \tfrac{109093 y^{2}}{1125} + \tfrac{136859 y z^{2}}{375} - \tfrac{519686 y z}{1125} + \tfrac{106859 y}{1125} + \tfrac{431 z^{3}}{3} - 321 z^{2} + 210 z - \tfrac{95}{3}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle \begin{cases} 45 x^{2} z + \tfrac{29717 x y z}{225} + \tfrac{39791 x z^{2}}{450} - 49 x z - x + 20 y^{2} z - \tfrac{16217 y z^{2}}{450} - 20 y z - y - \tfrac{25079 z^{3}}{225} + \tfrac{61 z^{2}}{2} - z + 1&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\45 x^{2} y + \tfrac{133909 x y^{2}}{450} - \tfrac{17342 x y z}{225} - 49 x y - x - \tfrac{97217 y^{3}}{450} + \tfrac{15421 y^{2} z}{225} + \tfrac{61 y^{2}}{2} + 20 y z^{2} - 20 y z - y - z + 1&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{101 x^{3}}{2} + \tfrac{34217 x^{2} y}{450} - \tfrac{6421 x^{2} z}{225} - \tfrac{37 x^{2}}{2} - x + 20 y^{2} z + 20 y z^{2} - 20 y z - y - z + 1&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{641 x^{3}}{2} + \tfrac{107683 x^{2} y}{150} + \tfrac{77371 x^{2} z}{75} - \tfrac{1573 x^{2}}{2} + \tfrac{243607 x y^{2}}{450} + 1541 x y z - \tfrac{260291 x y}{225} + \tfrac{431843 x z^{2}}{450} - \tfrac{354409 x z}{225} + \tfrac{1229 x}{2} + \tfrac{64783 y^{3}}{450} + \tfrac{129704 y^{2} z}{225} - \tfrac{196841 y^{2}}{450} + \tfrac{306467 y z^{2}}{450} - 1124 y z + \tfrac{198883 y}{450} + \tfrac{55921 z^{3}}{225} - \tfrac{290959 z^{2}}{450} + \tfrac{122971 z}{225} - \tfrac{297}{2}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(- 68434 x y + 7217 x z - 38250 y^{2} - 37091 y z + 36450 y + 45658 z^{2} - 15525 z + 450\right)}{450}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 61217 x y - 29025 y^{2} - 658 y z + 20925 y + 450\right)}{450}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(43217 x^{2} - 95209 x y + 9658 x z - 15525 x - 38250 y^{2} - 10316 y z + 36450 y + 450\right)}{450}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{28783 x^{3}}{450} - \tfrac{51907 x^{2} y}{450} - \tfrac{41954 x^{2} z}{225} + \tfrac{78041 x^{2}}{450} - \tfrac{15883 x y^{2}}{150} - 349 x y z + \tfrac{59591 x y}{225} - \tfrac{81467 x z^{2}}{450} + 336 x z - \tfrac{70183 x}{450} - \tfrac{109 y^{3}}{2} - \tfrac{17221 y^{2} z}{75} + \tfrac{317 y^{2}}{2} - \tfrac{105143 y z^{2}}{450} + \tfrac{86209 y z}{225} - \tfrac{301 y}{2} - \tfrac{13171 z^{3}}{225} + \tfrac{73159 z^{2}}{450} - \tfrac{33871 z}{225} + \tfrac{93}{2}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(13967 x z + 8533 y z - 900 z^{2} - 7200 z + 450\right)}{450}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 67967 x y + 81934 x z + 36467 y^{2} - 46834 y z + 12600 y + 18000 z^{2} - 19800 z + 450\right)}{450}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(- 31967 x^{2} + 467 x y + 27034 x z + 12600 x + 8066 y z + 18000 z^{2} - 19800 z + 450\right)}{450}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{103967 x^{3}}{450} - \tfrac{243467 x^{2} y}{450} - \tfrac{190384 x^{2} z}{225} + \tfrac{128267 x^{2}}{225} - \tfrac{175033 x y^{2}}{450} - 1224 x y z + 836 x y - \tfrac{128567 x z^{2}}{150} + \tfrac{294334 x z}{225} - \tfrac{201617 x}{450} - \tfrac{35533 y^{3}}{450} - \tfrac{85016 y^{2} z}{225} + \tfrac{59833 y^{2}}{225} - \tfrac{81133 y z^{2}}{150} + \tfrac{188966 y z}{225} - \tfrac{133183 y}{450} - 242 z^{3} + 596 z^{2} - 463 z + 109&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle \begin{cases} 45 x^{2} z + \tfrac{29717 x y z}{225} + \tfrac{39791 x z^{2}}{450} - 49 x z + 20 y^{2} z - \tfrac{16217 y z^{2}}{450} - 20 y z + y - \tfrac{25079 z^{3}}{225} + \tfrac{61 z^{2}}{2}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(20250 x^{2} + 133909 x y - 34684 x z - 22050 x - 97217 y^{2} + 30842 y z + 13725 y + 9000 z^{2} - 9000 z + 450\right)}{450}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{101 x^{3}}{2} + \tfrac{34217 x^{2} y}{450} - \tfrac{6421 x^{2} z}{225} - \tfrac{37 x^{2}}{2} + 20 y^{2} z + 20 y z^{2} - 20 y z + y&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{641 x^{3}}{2} + \tfrac{107683 x^{2} y}{150} + \tfrac{77371 x^{2} z}{75} - \tfrac{1573 x^{2}}{2} + \tfrac{243607 x y^{2}}{450} + 1541 x y z - \tfrac{260291 x y}{225} + \tfrac{431843 x z^{2}}{450} - \tfrac{354409 x z}{225} + \tfrac{1231 x}{2} + \tfrac{64783 y^{3}}{450} + \tfrac{129704 y^{2} z}{225} - \tfrac{196841 y^{2}}{450} + \tfrac{306467 y z^{2}}{450} - 1124 y z + \tfrac{199783 y}{450} + \tfrac{55921 z^{3}}{225} - \tfrac{290959 z^{2}}{450} + \tfrac{123196 z}{225} - \tfrac{299}{2}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} - \tfrac{34217 x y z}{225} + \tfrac{7217 x z^{2}}{450} - 85 y^{2} z - \tfrac{37091 y z^{2}}{450} + 81 y z - y + \tfrac{22829 z^{3}}{225} - \tfrac{69 z^{2}}{2} + 2 z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 61217 x y - 29025 y^{2} - 658 y z + 20925 y + 450\right)}{450}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{43217 x^{3}}{450} - \tfrac{95209 x^{2} y}{450} + \tfrac{4829 x^{2} z}{225} - \tfrac{69 x^{2}}{2} - 85 x y^{2} - \tfrac{5158 x y z}{225} + 81 x y + 2 x - y&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{28783 x^{3}}{450} - \tfrac{51907 x^{2} y}{450} - \tfrac{41954 x^{2} z}{225} + \tfrac{78041 x^{2}}{450} - \tfrac{15883 x y^{2}}{150} - 349 x y z + \tfrac{59591 x y}{225} - \tfrac{81467 x z^{2}}{450} + 336 x z - \tfrac{70633 x}{450} - \tfrac{109 y^{3}}{2} - \tfrac{17221 y^{2} z}{75} + \tfrac{317 y^{2}}{2} - \tfrac{105143 y z^{2}}{450} + \tfrac{86209 y z}{225} - \tfrac{305 y}{2} - \tfrac{13171 z^{3}}{225} + \tfrac{73159 z^{2}}{450} - \tfrac{34096 z}{225} + \tfrac{95}{2}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(13967 x z + 8533 y z - 900 z^{2} - 7200 z + 450\right)}{450}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 67967 x y + 81934 x z + 36467 y^{2} - 46834 y z + 12600 y + 18000 z^{2} - 19800 z + 450\right)}{450}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(- 31967 x^{2} + 467 x y + 27034 x z + 12600 x + 8066 y z + 18000 z^{2} - 19800 z + 450\right)}{450}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{103967 x^{3}}{450} - \tfrac{243467 x^{2} y}{450} - \tfrac{190384 x^{2} z}{225} + \tfrac{128267 x^{2}}{225} - \tfrac{175033 x y^{2}}{450} - 1224 x y z + 836 x y - \tfrac{128567 x z^{2}}{150} + \tfrac{294334 x z}{225} - \tfrac{201617 x}{450} - \tfrac{35533 y^{3}}{450} - \tfrac{85016 y^{2} z}{225} + \tfrac{59833 y^{2}}{225} - \tfrac{81133 y z^{2}}{150} + \tfrac{188966 y z}{225} - \tfrac{133183 y}{450} - 242 z^{3} + 596 z^{2} - 463 z + 109&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,1)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle \begin{cases} \tfrac{z \left(20250 x^{2} + 59434 x y + 39791 x z - 22050 x + 9000 y^{2} - 16217 y z - 9000 y - 50158 z^{2} + 13725 z + 450\right)}{450}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\45 x^{2} y + \tfrac{133909 x y^{2}}{450} - \tfrac{17342 x y z}{225} - 49 x y - \tfrac{97217 y^{3}}{450} + \tfrac{15421 y^{2} z}{225} + \tfrac{61 y^{2}}{2} + 20 y z^{2} - 20 y z + z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{101 x^{3}}{2} + \tfrac{34217 x^{2} y}{450} - \tfrac{6421 x^{2} z}{225} - \tfrac{37 x^{2}}{2} + 20 y^{2} z + 20 y z^{2} - 20 y z + z&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{641 x^{3}}{2} + \tfrac{107683 x^{2} y}{150} + \tfrac{77371 x^{2} z}{75} - \tfrac{1573 x^{2}}{2} + \tfrac{243607 x y^{2}}{450} + 1541 x y z - \tfrac{260291 x y}{225} + \tfrac{431843 x z^{2}}{450} - \tfrac{354409 x z}{225} + \tfrac{1231 x}{2} + \tfrac{64783 y^{3}}{450} + \tfrac{129704 y^{2} z}{225} - \tfrac{196841 y^{2}}{450} + \tfrac{306467 y z^{2}}{450} - 1124 y z + \tfrac{199333 y}{450} + \tfrac{55921 z^{3}}{225} - \tfrac{290959 z^{2}}{450} + \tfrac{123421 z}{225} - \tfrac{299}{2}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(- 68434 x y + 7217 x z - 38250 y^{2} - 37091 y z + 36450 y + 45658 z^{2} - 15525 z + 450\right)}{450}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 61217 x y - 29025 y^{2} - 658 y z + 20925 y + 450\right)}{450}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(43217 x^{2} - 95209 x y + 9658 x z - 15525 x - 38250 y^{2} - 10316 y z + 36450 y + 450\right)}{450}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{28783 x^{3}}{450} - \tfrac{51907 x^{2} y}{450} - \tfrac{41954 x^{2} z}{225} + \tfrac{78041 x^{2}}{450} - \tfrac{15883 x y^{2}}{150} - 349 x y z + \tfrac{59591 x y}{225} - \tfrac{81467 x z^{2}}{450} + 336 x z - \tfrac{70183 x}{450} - \tfrac{109 y^{3}}{2} - \tfrac{17221 y^{2} z}{75} + \tfrac{317 y^{2}}{2} - \tfrac{105143 y z^{2}}{450} + \tfrac{86209 y z}{225} - \tfrac{301 y}{2} - \tfrac{13171 z^{3}}{225} + \tfrac{73159 z^{2}}{450} - \tfrac{33871 z}{225} + \tfrac{93}{2}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(13967 x z + 8533 y z - 900 z^{2} - 7200 z + 450\right)}{450}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\- \tfrac{67967 x y^{2}}{450} + \tfrac{40967 x y z}{225} + \tfrac{36467 y^{3}}{450} - \tfrac{23417 y^{2} z}{225} + 28 y^{2} + 40 y z^{2} - 44 y z + 2 y - z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{31967 x^{3}}{450} + \tfrac{467 x^{2} y}{450} + \tfrac{13517 x^{2} z}{225} + 28 x^{2} + \tfrac{4033 x y z}{225} + 40 x z^{2} - 44 x z + 2 x - z&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{103967 x^{3}}{450} - \tfrac{243467 x^{2} y}{450} - \tfrac{190384 x^{2} z}{225} + \tfrac{128267 x^{2}}{225} - \tfrac{175033 x y^{2}}{450} - 1224 x y z + 836 x y - \tfrac{128567 x z^{2}}{150} + \tfrac{294334 x z}{225} - \tfrac{202067 x}{450} - \tfrac{35533 y^{3}}{450} - \tfrac{85016 y^{2} z}{225} + \tfrac{59833 y^{2}}{225} - \tfrac{81133 y z^{2}}{150} + \tfrac{188966 y z}{225} - \tfrac{133633 y}{450} - 242 z^{3} + 596 z^{2} - 465 z + 110&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,0)\cdot\left(\begin{array}{c}0\\-1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle \begin{cases} z \left(- 40 x^{2} - 72 x y - 6 x z + 44 x + 96 y z - 26 z^{2} - 28 z - 1\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\y \left(- 40 x^{2} + 50 x y - 128 x z + 44 x + 16 y^{2} + 54 y z - 28 y - 1\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\x \left(2 x^{2} + 24 x y - 74 x z + 16 x - 1\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\242 x^{3} + 692 x^{2} y + 706 x^{2} z - 596 x^{2} + 626 x y^{2} + 1224 x y z - 1088 x y + 598 x z^{2} - 1060 x z + 463 x + 176 y^{3} + 486 y^{2} z - 460 y^{2} + 444 y z^{2} - 836 y z + 393 y + 134 z^{3} - 376 z^{2} + 351 z - 109&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} - 20 x^{2} z + \tfrac{24262 x y z}{1125} - \tfrac{45881 x z^{2}}{1125} + 20 x z + x + \tfrac{10762 y^{2} z}{1125} - \tfrac{39619 y z^{2}}{375} - \tfrac{6262 y z}{1125} + y + \tfrac{119738 z^{3}}{1125} - \tfrac{3619 z^{2}}{1125} + z - 1&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\- 20 x^{2} z + \tfrac{881 x y^{2}}{1125} - 20 x z^{2} + 20 x z + x - \tfrac{26119 y^{3}}{1125} + \tfrac{37762 y^{2} z}{1125} - \tfrac{9881 y^{2}}{1125} + y + z - 1&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{156619 x^{3}}{1125} - \tfrac{192619 x^{2} y}{1125} - \tfrac{82762 x^{2} z}{1125} - \tfrac{3619 x^{2}}{1125} + \tfrac{10762 x y^{2}}{1125} + \tfrac{98024 x y z}{1125} - \tfrac{6262 x y}{1125} - 20 x z^{2} + 20 x z + x + y + z - 1&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{248381 x^{3}}{1125} - \tfrac{197081 x^{2} y}{225} - \tfrac{827024 x^{2} z}{1125} + \tfrac{231881 x^{2}}{375} - \tfrac{1189667 x y^{2}}{1125} - \tfrac{2163334 x y z}{1125} + \tfrac{366262 x y}{225} - \tfrac{172781 x z^{2}}{225} + \tfrac{1510048 x z}{1125} - \tfrac{215006 x}{375} - \tfrac{150881 y^{3}}{375} - \tfrac{260062 y^{2} z}{225} + \tfrac{1099667 y^{2}}{1125} - \tfrac{125881 y z^{2}}{125} + \tfrac{1978834 y z}{1125} - \tfrac{168956 y}{225} - \tfrac{285262 z^{3}}{1125} + \tfrac{153881 z^{2}}{225} - \tfrac{681899 z}{1125} + \tfrac{197756}{1125}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(32869 x z + 29738 y z + 41869 z^{2} - 21619 z - 1125\right)}{1125}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 55369 x y + 88238 x z + 15262 y^{2} + 22107 y z + 8119 y + 34238 z^{2} - 29738 z - 1125\right)}{1125}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(- 145369 x^{2} + 105262 x y + 185869 x z + 8119 x - 75524 y z + 34238 z^{2} - 29738 z - 1125\right)}{1125}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{34631 x^{3}}{1125} + \tfrac{264524 x^{2} y}{1125} + \tfrac{33731 x^{2} z}{225} - \tfrac{50381 x^{2}}{375} + \tfrac{85031 x y^{2}}{225} + \tfrac{822334 x y z}{1125} - \tfrac{623548 x y}{1125} + \tfrac{287417 x z^{2}}{1125} - \tfrac{98062 x z}{225} + \tfrac{66506 x}{375} + \tfrac{195262 y^{3}}{1125} + \tfrac{72631 y^{2} z}{125} - \tfrac{94481 y^{2}}{225} + \tfrac{122362 y z^{2}}{225} - \tfrac{975334 y z}{1125} + \tfrac{360149 y}{1125} + \tfrac{51131 z^{3}}{375} - \tfrac{393167 z^{2}}{1125} + \tfrac{64556 z}{225} - \tfrac{83006}{1125}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,0)\cdot\left(\begin{array}{c}0\\-1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle \begin{cases} - 40 x^{2} z - 72 x y z - 6 x z^{2} + 44 x z + x + 96 y z^{2} - 26 z^{3} - 28 z^{2} - 2 z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\- 40 x^{2} y + 50 x y^{2} - 128 x y z + 44 x y + x + 16 y^{3} + 54 y^{2} z - 28 y^{2} - 2 y&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\x \left(2 x^{2} + 24 x y - 74 x z + 16 x - 1\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\242 x^{3} + 692 x^{2} y + 706 x^{2} z - 596 x^{2} + 626 x y^{2} + 1224 x y z - 1088 x y + 598 x z^{2} - 1060 x z + 465 x + 176 y^{3} + 486 y^{2} z - 460 y^{2} + 444 y z^{2} - 836 y z + 394 y + 134 z^{3} - 376 z^{2} + 352 z - 110&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} - 20 x^{2} z + \tfrac{24262 x y z}{1125} - \tfrac{45881 x z^{2}}{1125} + 20 x z - x + \tfrac{10762 y^{2} z}{1125} - \tfrac{39619 y z^{2}}{375} - \tfrac{6262 y z}{1125} + \tfrac{119738 z^{3}}{1125} - \tfrac{3619 z^{2}}{1125}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\- 20 x^{2} z + \tfrac{881 x y^{2}}{1125} - 20 x z^{2} + 20 x z - x - \tfrac{26119 y^{3}}{1125} + \tfrac{37762 y^{2} z}{1125} - \tfrac{9881 y^{2}}{1125}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(156619 x^{2} - 192619 x y - 82762 x z - 3619 x + 10762 y^{2} + 98024 y z - 6262 y - 22500 z^{2} + 22500 z - 1125\right)}{1125}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{248381 x^{3}}{1125} - \tfrac{197081 x^{2} y}{225} - \tfrac{827024 x^{2} z}{1125} + \tfrac{231881 x^{2}}{375} - \tfrac{1189667 x y^{2}}{1125} - \tfrac{2163334 x y z}{1125} + \tfrac{366262 x y}{225} - \tfrac{172781 x z^{2}}{225} + \tfrac{1510048 x z}{1125} - \tfrac{215756 x}{375} - \tfrac{150881 y^{3}}{375} - \tfrac{260062 y^{2} z}{225} + \tfrac{1099667 y^{2}}{1125} - \tfrac{125881 y z^{2}}{125} + \tfrac{1978834 y z}{1125} - \tfrac{169181 y}{225} - \tfrac{285262 z^{3}}{1125} + \tfrac{153881 z^{2}}{225} - \tfrac{683024 z}{1125} + \tfrac{198881}{1125}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(32869 x z + 29738 y z + 41869 z^{2} - 21619 z - 1125\right)}{1125}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 55369 x y + 88238 x z + 15262 y^{2} + 22107 y z + 8119 y + 34238 z^{2} - 29738 z - 1125\right)}{1125}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(- 145369 x^{2} + 105262 x y + 185869 x z + 8119 x - 75524 y z + 34238 z^{2} - 29738 z - 1125\right)}{1125}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{34631 x^{3}}{1125} + \tfrac{264524 x^{2} y}{1125} + \tfrac{33731 x^{2} z}{225} - \tfrac{50381 x^{2}}{375} + \tfrac{85031 x y^{2}}{225} + \tfrac{822334 x y z}{1125} - \tfrac{623548 x y}{1125} + \tfrac{287417 x z^{2}}{1125} - \tfrac{98062 x z}{225} + \tfrac{66506 x}{375} + \tfrac{195262 y^{3}}{1125} + \tfrac{72631 y^{2} z}{125} - \tfrac{94481 y^{2}}{225} + \tfrac{122362 y z^{2}}{225} - \tfrac{975334 y z}{1125} + \tfrac{360149 y}{1125} + \tfrac{51131 z^{3}}{375} - \tfrac{393167 z^{2}}{1125} + \tfrac{64556 z}{225} - \tfrac{83006}{1125}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,1)\cdot\left(\begin{array}{c}0\\-1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle \begin{cases} z \left(- 40 x^{2} - 72 x y - 6 x z + 44 x + 96 y z - 26 z^{2} - 28 z - 1\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\y \left(- 40 x^{2} + 50 x y - 128 x z + 44 x + 16 y^{2} + 54 y z - 28 y - 1\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\x \left(2 x^{2} + 24 x y - 74 x z + 16 x - 1\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\242 x^{3} + 692 x^{2} y + 706 x^{2} z - 596 x^{2} + 626 x y^{2} + 1224 x y z - 1088 x y + 598 x z^{2} - 1060 x z + 463 x + 176 y^{3} + 486 y^{2} z - 460 y^{2} + 444 y z^{2} - 836 y z + 393 y + 134 z^{3} - 376 z^{2} + 351 z - 109&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(- 22500 x^{2} + 24262 x y - 45881 x z + 22500 x + 10762 y^{2} - 118857 y z - 6262 y + 119738 z^{2} - 3619 z - 1125\right)}{1125}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\- 20 x^{2} z + \tfrac{881 x y^{2}}{1125} - 20 x z^{2} + 20 x z - \tfrac{26119 y^{3}}{1125} + \tfrac{37762 y^{2} z}{1125} - \tfrac{9881 y^{2}}{1125} - z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{156619 x^{3}}{1125} - \tfrac{192619 x^{2} y}{1125} - \tfrac{82762 x^{2} z}{1125} - \tfrac{3619 x^{2}}{1125} + \tfrac{10762 x y^{2}}{1125} + \tfrac{98024 x y z}{1125} - \tfrac{6262 x y}{1125} - 20 x z^{2} + 20 x z - z&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{248381 x^{3}}{1125} - \tfrac{197081 x^{2} y}{225} - \tfrac{827024 x^{2} z}{1125} + \tfrac{231881 x^{2}}{375} - \tfrac{1189667 x y^{2}}{1125} - \tfrac{2163334 x y z}{1125} + \tfrac{366262 x y}{225} - \tfrac{172781 x z^{2}}{225} + \tfrac{1510048 x z}{1125} - \tfrac{215381 x}{375} - \tfrac{150881 y^{3}}{375} - \tfrac{260062 y^{2} z}{225} + \tfrac{1099667 y^{2}}{1125} - \tfrac{125881 y z^{2}}{125} + \tfrac{1978834 y z}{1125} - \tfrac{169181 y}{225} - \tfrac{285262 z^{3}}{1125} + \tfrac{153881 z^{2}}{225} - \tfrac{684149 z}{1125} + \tfrac{198881}{1125}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(32869 x z + 29738 y z + 41869 z^{2} - 21619 z - 1125\right)}{1125}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\- \tfrac{55369 x y^{2}}{1125} + \tfrac{88238 x y z}{1125} + \tfrac{15262 y^{3}}{1125} + \tfrac{7369 y^{2} z}{375} + \tfrac{8119 y^{2}}{1125} + \tfrac{34238 y z^{2}}{1125} - \tfrac{29738 y z}{1125} - 2 y + z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{145369 x^{3}}{1125} + \tfrac{105262 x^{2} y}{1125} + \tfrac{185869 x^{2} z}{1125} + \tfrac{8119 x^{2}}{1125} - \tfrac{75524 x y z}{1125} + \tfrac{34238 x z^{2}}{1125} - \tfrac{29738 x z}{1125} - 2 x + z&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{34631 x^{3}}{1125} + \tfrac{264524 x^{2} y}{1125} + \tfrac{33731 x^{2} z}{225} - \tfrac{50381 x^{2}}{375} + \tfrac{85031 x y^{2}}{225} + \tfrac{822334 x y z}{1125} - \tfrac{623548 x y}{1125} + \tfrac{287417 x z^{2}}{1125} - \tfrac{98062 x z}{225} + \tfrac{66881 x}{375} + \tfrac{195262 y^{3}}{1125} + \tfrac{72631 y^{2} z}{125} - \tfrac{94481 y^{2}}{225} + \tfrac{122362 y z^{2}}{225} - \tfrac{975334 y z}{1125} + \tfrac{361274 y}{1125} + \tfrac{51131 z^{3}}{375} - \tfrac{393167 z^{2}}{1125} + \tfrac{65006 z}{225} - \tfrac{84131}{1125}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{9}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle \begin{cases} \tfrac{z \left(- 343124 x^{2} - 421874 x y - 485436 x z + 334124 x + 233437 y z + 258749 z^{2} - 153562 z + 2250\right)}{2250}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 343124 x^{2} - 479812 x y - 427498 x z + 334124 x + 413437 y^{2} + 78749 y z - 153562 y + 2250\right)}{2250}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(- 221062 x^{2} - 188437 x y - 348749 x z + 180562 x + 2250\right)}{2250}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{17531 x^{3}}{375} + \tfrac{342559 x^{2} y}{2250} + \tfrac{38699 x^{2} z}{450} - \tfrac{87467 x^{2}}{1125} + \tfrac{29081 x y^{2}}{225} + \tfrac{138934 x y z}{1125} - \tfrac{33412 x y}{225} - \tfrac{719 x z^{2}}{125} - \tfrac{15184 x z}{1125} + \tfrac{4106 x}{225} + \tfrac{53437 y^{3}}{2250} + \tfrac{5623 y^{2} z}{2250} - \tfrac{13406 y^{2}}{375} - \tfrac{29813 y z^{2}}{450} + \tfrac{74252 y z}{1125} - \tfrac{563 y}{750} - \tfrac{101251 z^{3}}{2250} + \tfrac{22894 z^{2}}{225} - \tfrac{156377 z}{2250} + \tfrac{14344}{1125}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(230 x y - 95 x z + 80 y^{2} - 74 y z - 88 y - 45 z^{2} + 56 z + 2\right)}{2}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(135 x y - 4 y^{2} - 35 y z - 32 y + 2\right)}{2}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(65 x^{2} - 14 x y - 205 x z + 56 x + 80 y^{2} + 170 y z - 88 y + 2\right)}{2}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{255 x^{3}}{2} - 572 x^{2} y - \tfrac{875 x^{2} z}{2} + 363 x^{2} - \tfrac{1373 x y^{2}}{2} - 1224 x y z + 1034 x y - \tfrac{985 x z^{2}}{2} + 836 x z - \tfrac{689 x}{2} - 242 y^{3} - \tfrac{1423 y^{2} z}{2} + 596 y^{2} - 652 y z^{2} + 1114 y z - 463 y - \tfrac{365 z^{3}}{2} + 473 z^{2} - \tfrac{799 z}{2} + 109&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 20 x^{2} y + 20 x y^{2} - 20 x y + \tfrac{213749 x z^{2}}{2250} - x + \tfrac{120937 y z^{2}}{2250} - y + \tfrac{94781 z^{3}}{1125} - \tfrac{58781 z^{2}}{1125} - z + 1&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\20 x^{2} y - \tfrac{123749 x y^{2}}{2250} + \tfrac{191249 x y z}{1125} - 20 x y - x - \tfrac{435937 y^{3}}{2250} + \tfrac{246656 y^{2} z}{1125} + \tfrac{72281 y^{2}}{1125} + \tfrac{126562 y z^{2}}{1125} - \tfrac{131062 y z}{1125} - y - z + 1&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{528749 x^{3}}{2250} - \tfrac{30937 x^{2} y}{2250} + \tfrac{113156 x^{2} z}{375} + \tfrac{72281 x^{2}}{1125} + 20 x y^{2} + \tfrac{98437 x y z}{1125} - 20 x y + \tfrac{126562 x z^{2}}{1125} - \tfrac{131062 x z}{1125} - x - y - z + 1&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{281251 x^{3}}{2250} + \tfrac{205313 x^{2} y}{450} + \tfrac{53969 x^{2} z}{125} - \tfrac{82294 x^{2}}{225} + \tfrac{1119377 x y^{2}}{2250} + \tfrac{1202066 x y z}{1125} - \tfrac{960752 x y}{1125} + \tfrac{236701 x z^{2}}{450} - \tfrac{988316 x z}{1125} + \tfrac{799877 x}{2250} + \tfrac{374063 y^{3}}{2250} + \tfrac{134269 y^{2} z}{225} - \tfrac{168094 y^{2}}{375} + \tfrac{1461941 y z^{2}}{2250} - \tfrac{234788 y z}{225} + \tfrac{297563 y}{750} + \tfrac{82219 z^{3}}{375} - \tfrac{619033 z^{2}}{1125} + \tfrac{100294 z}{225} - \tfrac{129094}{1125}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{10}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle \begin{cases} - \tfrac{171562 x^{2} z}{1125} - \tfrac{210937 x y z}{1125} - \tfrac{80906 x z^{2}}{375} + \tfrac{167062 x z}{1125} - x + \tfrac{233437 y z^{2}}{2250} + \tfrac{258749 z^{3}}{2250} - \tfrac{76781 z^{2}}{1125} + 2 z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\- \tfrac{171562 x^{2} y}{1125} - \tfrac{239906 x y^{2}}{1125} - \tfrac{213749 x y z}{1125} + \tfrac{167062 x y}{1125} - x + \tfrac{413437 y^{3}}{2250} + \tfrac{78749 y^{2} z}{2250} - \tfrac{76781 y^{2}}{1125} + 2 y&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(- 221062 x^{2} - 188437 x y - 348749 x z + 180562 x + 2250\right)}{2250}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{17531 x^{3}}{375} + \tfrac{342559 x^{2} y}{2250} + \tfrac{38699 x^{2} z}{450} - \tfrac{87467 x^{2}}{1125} + \tfrac{29081 x y^{2}}{225} + \tfrac{138934 x y z}{1125} - \tfrac{33412 x y}{225} - \tfrac{719 x z^{2}}{125} - \tfrac{15184 x z}{1125} + \tfrac{3656 x}{225} + \tfrac{53437 y^{3}}{2250} + \tfrac{5623 y^{2} z}{2250} - \tfrac{13406 y^{2}}{375} - \tfrac{29813 y z^{2}}{450} + \tfrac{74252 y z}{1125} - \tfrac{1313 y}{750} - \tfrac{101251 z^{3}}{2250} + \tfrac{22894 z^{2}}{225} - \tfrac{158627 z}{2250} + \tfrac{15469}{1125}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(230 x y - 95 x z + 80 y^{2} - 74 y z - 88 y - 45 z^{2} + 56 z + 2\right)}{2}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(135 x y - 4 y^{2} - 35 y z - 32 y + 2\right)}{2}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(65 x^{2} - 14 x y - 205 x z + 56 x + 80 y^{2} + 170 y z - 88 y + 2\right)}{2}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{255 x^{3}}{2} - 572 x^{2} y - \tfrac{875 x^{2} z}{2} + 363 x^{2} - \tfrac{1373 x y^{2}}{2} - 1224 x y z + 1034 x y - \tfrac{985 x z^{2}}{2} + 836 x z - \tfrac{689 x}{2} - 242 y^{3} - \tfrac{1423 y^{2} z}{2} + 596 y^{2} - 652 y z^{2} + 1114 y z - 463 y - \tfrac{365 z^{3}}{2} + 473 z^{2} - \tfrac{799 z}{2} + 109&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 20 x^{2} y + 20 x y^{2} - 20 x y + \tfrac{213749 x z^{2}}{2250} + x + \tfrac{120937 y z^{2}}{2250} + \tfrac{94781 z^{3}}{1125} - \tfrac{58781 z^{2}}{1125}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\20 x^{2} y - \tfrac{123749 x y^{2}}{2250} + \tfrac{191249 x y z}{1125} - 20 x y + x - \tfrac{435937 y^{3}}{2250} + \tfrac{246656 y^{2} z}{1125} + \tfrac{72281 y^{2}}{1125} + \tfrac{126562 y z^{2}}{1125} - \tfrac{131062 y z}{1125}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(- 528749 x^{2} - 30937 x y + 678936 x z + 144562 x + 45000 y^{2} + 196874 y z - 45000 y + 253124 z^{2} - 262124 z + 2250\right)}{2250}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{281251 x^{3}}{2250} + \tfrac{205313 x^{2} y}{450} + \tfrac{53969 x^{2} z}{125} - \tfrac{82294 x^{2}}{225} + \tfrac{1119377 x y^{2}}{2250} + \tfrac{1202066 x y z}{1125} - \tfrac{960752 x y}{1125} + \tfrac{236701 x z^{2}}{450} - \tfrac{988316 x z}{1125} + \tfrac{804377 x}{2250} + \tfrac{374063 y^{3}}{2250} + \tfrac{134269 y^{2} z}{225} - \tfrac{168094 y^{2}}{375} + \tfrac{1461941 y z^{2}}{2250} - \tfrac{234788 y z}{225} + \tfrac{298313 y}{750} + \tfrac{82219 z^{3}}{375} - \tfrac{619033 z^{2}}{1125} + \tfrac{100519 z}{225} - \tfrac{130219}{1125}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{11}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle \begin{cases} \tfrac{z \left(- 343124 x^{2} - 421874 x y - 485436 x z + 334124 x + 233437 y z + 258749 z^{2} - 153562 z + 2250\right)}{2250}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 343124 x^{2} - 479812 x y - 427498 x z + 334124 x + 413437 y^{2} + 78749 y z - 153562 y + 2250\right)}{2250}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(- 221062 x^{2} - 188437 x y - 348749 x z + 180562 x + 2250\right)}{2250}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{17531 x^{3}}{375} + \tfrac{342559 x^{2} y}{2250} + \tfrac{38699 x^{2} z}{450} - \tfrac{87467 x^{2}}{1125} + \tfrac{29081 x y^{2}}{225} + \tfrac{138934 x y z}{1125} - \tfrac{33412 x y}{225} - \tfrac{719 x z^{2}}{125} - \tfrac{15184 x z}{1125} + \tfrac{4106 x}{225} + \tfrac{53437 y^{3}}{2250} + \tfrac{5623 y^{2} z}{2250} - \tfrac{13406 y^{2}}{375} - \tfrac{29813 y z^{2}}{450} + \tfrac{74252 y z}{1125} - \tfrac{563 y}{750} - \tfrac{101251 z^{3}}{2250} + \tfrac{22894 z^{2}}{225} - \tfrac{156377 z}{2250} + \tfrac{14344}{1125}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 115 x y z - \tfrac{95 x z^{2}}{2} + 40 y^{2} z - 37 y z^{2} - 44 y z - y - \tfrac{45 z^{3}}{2} + 28 z^{2} + 2 z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(135 x y - 4 y^{2} - 35 y z - 32 y + 2\right)}{2}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{65 x^{3}}{2} - 7 x^{2} y - \tfrac{205 x^{2} z}{2} + 28 x^{2} + 40 x y^{2} + 85 x y z - 44 x y + 2 x - y&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{255 x^{3}}{2} - 572 x^{2} y - \tfrac{875 x^{2} z}{2} + 363 x^{2} - \tfrac{1373 x y^{2}}{2} - 1224 x y z + 1034 x y - \tfrac{985 x z^{2}}{2} + 836 x z - \tfrac{691 x}{2} - 242 y^{3} - \tfrac{1423 y^{2} z}{2} + 596 y^{2} - 652 y z^{2} + 1114 y z - 465 y - \tfrac{365 z^{3}}{2} + 473 z^{2} - \tfrac{801 z}{2} + 110&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 20 x^{2} y + 20 x y^{2} - 20 x y + \tfrac{213749 x z^{2}}{2250} + \tfrac{120937 y z^{2}}{2250} + y + \tfrac{94781 z^{3}}{1125} - \tfrac{58781 z^{2}}{1125}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(45000 x^{2} - 123749 x y + 382498 x z - 45000 x - 435937 y^{2} + 493312 y z + 144562 y + 253124 z^{2} - 262124 z + 2250\right)}{2250}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{528749 x^{3}}{2250} - \tfrac{30937 x^{2} y}{2250} + \tfrac{113156 x^{2} z}{375} + \tfrac{72281 x^{2}}{1125} + 20 x y^{2} + \tfrac{98437 x y z}{1125} - 20 x y + \tfrac{126562 x z^{2}}{1125} - \tfrac{131062 x z}{1125} + y&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{281251 x^{3}}{2250} + \tfrac{205313 x^{2} y}{450} + \tfrac{53969 x^{2} z}{125} - \tfrac{82294 x^{2}}{225} + \tfrac{1119377 x y^{2}}{2250} + \tfrac{1202066 x y z}{1125} - \tfrac{960752 x y}{1125} + \tfrac{236701 x z^{2}}{450} - \tfrac{988316 x z}{1125} + \tfrac{802127 x}{2250} + \tfrac{374063 y^{3}}{2250} + \tfrac{134269 y^{2} z}{225} - \tfrac{168094 y^{2}}{375} + \tfrac{1461941 y z^{2}}{2250} - \tfrac{234788 y z}{225} + \tfrac{299063 y}{750} + \tfrac{82219 z^{3}}{375} - \tfrac{619033 z^{2}}{1125} + \tfrac{100519 z}{225} - \tfrac{130219}{1125}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{12}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{0}\)
where \(f_{0}\) is the 0th face;
and \(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0.

\(\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle \begin{cases} \tfrac{z \left(- 183750 x^{2} - 143936 x y - 222439 x z + 174750 x + 221968 y z + 428657 z^{2} - 208875 z + 2250\right)}{375}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 183750 x^{2} - 109061 x y - 257314 x z + 174750 x + 371968 y^{2} + 278657 y z - 208875 y + 2250\right)}{375}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(1125 x^{2} + 78032 x y + 21343 x z - 34125 x + 2250\right)}{375}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\263 x^{3} + \tfrac{91468 x^{2} y}{125} + \tfrac{148157 x^{2} z}{125} - 699 x^{2} + \tfrac{277747 x y^{2}}{375} + 2166 x y z - \tfrac{515122 x y}{375} + \tfrac{504503 x z^{2}}{375} - \tfrac{741878 x z}{375} + 627 x + \tfrac{101968 y^{3}}{375} + \tfrac{377593 y^{2} z}{375} - \tfrac{277811 y^{2}}{375} + \tfrac{434282 y z^{2}}{375} - 1824 y z + \tfrac{247468 y}{375} + \tfrac{158657 z^{3}}{375} - \tfrac{391189 z^{2}}{375} + \tfrac{304157 z}{375} - 191&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(- 163564 x y + 231782 x z - 183750 y^{2} - 202811 y z + 174750 y + 418843 z^{2} - 208875 z + 2250\right)}{375}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(68218 x y + 1125 y^{2} + 31157 y z - 34125 y + 2250\right)}{375}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(381782 x^{2} - 128689 x y + 268843 x z - 208875 x - 183750 y^{2} - 237686 y z + 174750 y + 2250\right)}{375}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{111782 x^{3}}{375} + \tfrac{317003 x^{2} y}{375} + \tfrac{387407 x^{2} z}{375} - \tfrac{297439 x^{2}}{375} + \tfrac{101282 x y^{2}}{125} + 2166 x y z - \tfrac{554378 x y}{375} + \tfrac{424468 x z^{2}}{375} - 1824 x z + \tfrac{257282 x}{375} + 263 y^{3} + \tfrac{138343 y^{2} z}{125} - 699 y^{2} + \tfrac{465247 y z^{2}}{375} - \tfrac{702622 y z}{375} + 627 y + \tfrac{148843 z^{3}}{375} - \tfrac{371561 z^{2}}{375} + \tfrac{294343 z}{375} - 191&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{2 z \left(132766 x z + 127859 y z + 70875 z^{2} - 87375 z + 1125\right)}{375}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{2 y \left(17234 x y + 115532 x z + 97141 y^{2} + 52843 y z - 34125 y + 48750 z^{2} - 53250 z + 1125\right)}{375}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{2 x \left(92234 x^{2} + 22141 x y + 62657 x z - 34125 x + 105718 y z + 48750 z^{2} - 53250 z + 1125\right)}{375}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{85532 x^{3}}{375} - \tfrac{231782 x^{2} y}{375} - \tfrac{612878 x^{2} z}{375} + \tfrac{237814 x^{2}}{375} - \tfrac{221968 x y^{2}}{375} - 3084 x y z + 1176 x y - \tfrac{283532 x z^{2}}{125} + \tfrac{1078628 x z}{375} - \tfrac{221282 x}{375} - \tfrac{75718 y^{3}}{375} - \tfrac{573622 y^{2} z}{375} + \tfrac{218186 y^{2}}{375} - \tfrac{273718 y z^{2}}{125} + \tfrac{1039372 y z}{375} - \tfrac{211468 y}{375} - 862 z^{3} + 1926 z^{2} - 1248 z + 184&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{1}\)
where \(f_{1}\) is the 1st face;
and \(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1.

\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle \begin{cases} \tfrac{z \left(- 20250 x^{2} - 59434 x y - 39791 x z + 22050 x - 9000 y^{2} + 16217 y z + 9000 y + 50158 z^{2} - 13725 z - 450\right)}{75}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 20250 x^{2} - 133909 x y + 34684 x z + 22050 x + 97217 y^{2} - 30842 y z - 13725 y - 9000 z^{2} + 9000 z - 450\right)}{75}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- 303 x^{3} - \tfrac{34217 x^{2} y}{75} + \tfrac{12842 x^{2} z}{75} + 111 x^{2} - 6 x - 120 y^{2} z - 120 y z^{2} + 120 y z&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- 1923 x^{3} - \tfrac{107683 x^{2} y}{25} - \tfrac{154742 x^{2} z}{25} + 4719 x^{2} - \tfrac{243607 x y^{2}}{75} - 9246 x y z + \tfrac{520582 x y}{75} - \tfrac{431843 x z^{2}}{75} + \tfrac{708818 x z}{75} - 3687 x - \tfrac{64783 y^{3}}{75} - \tfrac{259408 y^{2} z}{75} + \tfrac{196841 y^{2}}{75} - \tfrac{306467 y z^{2}}{75} + 6744 y z - \tfrac{198883 y}{75} - \tfrac{111842 z^{3}}{75} + \tfrac{290959 z^{2}}{75} - \tfrac{245942 z}{75} + 891&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(68434 x y - 7217 x z + 38250 y^{2} + 37091 y z - 36450 y - 45658 z^{2} + 15525 z - 450\right)}{75}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(61217 x y + 29025 y^{2} + 658 y z - 20925 y - 450\right)}{75}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(- 43217 x^{2} + 95209 x y - 9658 x z + 15525 x + 38250 y^{2} + 10316 y z - 36450 y - 450\right)}{75}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{28783 x^{3}}{75} + \tfrac{51907 x^{2} y}{75} + \tfrac{83908 x^{2} z}{75} - \tfrac{78041 x^{2}}{75} + \tfrac{15883 x y^{2}}{25} + 2094 x y z - \tfrac{119182 x y}{75} + \tfrac{81467 x z^{2}}{75} - 2016 x z + \tfrac{70183 x}{75} + 327 y^{3} + \tfrac{34442 y^{2} z}{25} - 951 y^{2} + \tfrac{105143 y z^{2}}{75} - \tfrac{172418 y z}{75} + 903 y + \tfrac{26342 z^{3}}{75} - \tfrac{73159 z^{2}}{75} + \tfrac{67742 z}{75} - 279&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(- 13967 x z - 8533 y z + 900 z^{2} + 7200 z - 450\right)}{75}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(67967 x y - 81934 x z - 36467 y^{2} + 46834 y z - 12600 y - 18000 z^{2} + 19800 z - 450\right)}{75}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(31967 x^{2} - 467 x y - 27034 x z - 12600 x - 8066 y z - 18000 z^{2} + 19800 z - 450\right)}{75}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{103967 x^{3}}{75} + \tfrac{243467 x^{2} y}{75} + \tfrac{380768 x^{2} z}{75} - \tfrac{256534 x^{2}}{75} + \tfrac{175033 x y^{2}}{75} + 7344 x y z - 5016 x y + \tfrac{128567 x z^{2}}{25} - \tfrac{588668 x z}{75} + \tfrac{201617 x}{75} + \tfrac{35533 y^{3}}{75} + \tfrac{170032 y^{2} z}{75} - \tfrac{119666 y^{2}}{75} + \tfrac{81133 y z^{2}}{25} - \tfrac{377932 y z}{75} + \tfrac{133183 y}{75} + 1452 z^{3} - 3576 z^{2} + 2778 z - 654&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{14}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{2}\)
where \(f_{2}\) is the 2nd face;
and \(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2.

\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle \begin{cases} 6 z \left(40 x^{2} + 72 x y + 6 x z - 44 x - 96 y z + 26 z^{2} + 28 z + 1\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\6 y \left(40 x^{2} - 50 x y + 128 x z - 44 x - 16 y^{2} - 54 y z + 28 y + 1\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\6 x \left(- 2 x^{2} - 24 x y + 74 x z - 16 x + 1\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- 1452 x^{3} - 4152 x^{2} y - 4236 x^{2} z + 3576 x^{2} - 3756 x y^{2} - 7344 x y z + 6528 x y - 3588 x z^{2} + 6360 x z - 2778 x - 1056 y^{3} - 2916 y^{2} z + 2760 y^{2} - 2664 y z^{2} + 5016 y z - 2358 y - 804 z^{3} + 2256 z^{2} - 2106 z + 654&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{2 z \left(22500 x^{2} - 24262 x y + 45881 x z - 22500 x - 10762 y^{2} + 118857 y z + 6262 y - 119738 z^{2} + 3619 z + 1125\right)}{375}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\120 x^{2} z - \tfrac{1762 x y^{2}}{375} + 120 x z^{2} - 120 x z + \tfrac{52238 y^{3}}{375} - \tfrac{75524 y^{2} z}{375} + \tfrac{19762 y^{2}}{375} + 6 y&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{2 x \left(- 156619 x^{2} + 192619 x y + 82762 x z + 3619 x - 10762 y^{2} - 98024 y z + 6262 y + 22500 z^{2} - 22500 z + 1125\right)}{375}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{496762 x^{3}}{375} + \tfrac{394162 x^{2} y}{75} + \tfrac{1654048 x^{2} z}{375} - \tfrac{463762 x^{2}}{125} + \tfrac{2379334 x y^{2}}{375} + \tfrac{4326668 x y z}{375} - \tfrac{732524 x y}{75} + \tfrac{345562 x z^{2}}{75} - \tfrac{3020096 x z}{375} + \tfrac{430012 x}{125} + \tfrac{301762 y^{3}}{125} + \tfrac{520124 y^{2} z}{75} - \tfrac{2199334 y^{2}}{375} + \tfrac{755286 y z^{2}}{125} - \tfrac{3957668 y z}{375} + \tfrac{337912 y}{75} + \tfrac{570524 z^{3}}{375} - \tfrac{307762 z^{2}}{75} + \tfrac{1363798 z}{375} - \tfrac{395512}{375}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{2 z \left(- 32869 x z - 29738 y z - 41869 z^{2} + 21619 z + 1125\right)}{375}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{2 y \left(55369 x y - 88238 x z - 15262 y^{2} - 22107 y z - 8119 y - 34238 z^{2} + 29738 z + 1125\right)}{375}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{2 x \left(145369 x^{2} - 105262 x y - 185869 x z - 8119 x + 75524 y z - 34238 z^{2} + 29738 z + 1125\right)}{375}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{69262 x^{3}}{375} - \tfrac{529048 x^{2} y}{375} - \tfrac{67462 x^{2} z}{75} + \tfrac{100762 x^{2}}{125} - \tfrac{170062 x y^{2}}{75} - \tfrac{1644668 x y z}{375} + \tfrac{1247096 x y}{375} - \tfrac{574834 x z^{2}}{375} + \tfrac{196124 x z}{75} - \tfrac{133012 x}{125} - \tfrac{390524 y^{3}}{375} - \tfrac{435786 y^{2} z}{125} + \tfrac{188962 y^{2}}{75} - \tfrac{244724 y z^{2}}{75} + \tfrac{1950668 y z}{375} - \tfrac{720298 y}{375} - \tfrac{102262 z^{3}}{125} + \tfrac{786334 z^{2}}{375} - \tfrac{129112 z}{75} + \tfrac{166012}{375}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{15}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{3}\)
where \(f_{3}\) is the 3th face;
and \(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3.

\(\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle \begin{cases} \tfrac{z \left(343124 x^{2} + 421874 x y + 485436 x z - 334124 x - 233437 y z - 258749 z^{2} + 153562 z - 2250\right)}{375}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(343124 x^{2} + 479812 x y + 427498 x z - 334124 x - 413437 y^{2} - 78749 y z + 153562 y - 2250\right)}{375}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(221062 x^{2} + 188437 x y + 348749 x z - 180562 x - 2250\right)}{375}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{35062 x^{3}}{125} - \tfrac{342559 x^{2} y}{375} - \tfrac{38699 x^{2} z}{75} + \tfrac{174934 x^{2}}{375} - \tfrac{58162 x y^{2}}{75} - \tfrac{277868 x y z}{375} + \tfrac{66824 x y}{75} + \tfrac{4314 x z^{2}}{125} + \tfrac{30368 x z}{375} - \tfrac{8212 x}{75} - \tfrac{53437 y^{3}}{375} - \tfrac{5623 y^{2} z}{375} + \tfrac{26812 y^{2}}{125} + \tfrac{29813 y z^{2}}{75} - \tfrac{148504 y z}{375} + \tfrac{563 y}{125} + \tfrac{101251 z^{3}}{375} - \tfrac{45788 z^{2}}{75} + \tfrac{156377 z}{375} - \tfrac{28688}{375}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 3 z \left(- 230 x y + 95 x z - 80 y^{2} + 74 y z + 88 y + 45 z^{2} - 56 z - 2\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\3 y \left(- 135 x y + 4 y^{2} + 35 y z + 32 y - 2\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\3 x \left(- 65 x^{2} + 14 x y + 205 x z - 56 x - 80 y^{2} - 170 y z + 88 y - 2\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\765 x^{3} + 3432 x^{2} y + 2625 x^{2} z - 2178 x^{2} + 4119 x y^{2} + 7344 x y z - 6204 x y + 2955 x z^{2} - 5016 x z + 2067 x + 1452 y^{3} + 4269 y^{2} z - 3576 y^{2} + 3912 y z^{2} - 6684 y z + 2778 y + 1095 z^{3} - 2838 z^{2} + 2397 z - 654&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} - 120 x^{2} y - 120 x y^{2} + 120 x y - \tfrac{213749 x z^{2}}{375} - \tfrac{120937 y z^{2}}{375} - \tfrac{189562 z^{3}}{375} + \tfrac{117562 z^{2}}{375} - 6 z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 45000 x^{2} + 123749 x y - 382498 x z + 45000 x + 435937 y^{2} - 493312 y z - 144562 y - 253124 z^{2} + 262124 z - 2250\right)}{375}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(528749 x^{2} + 30937 x y - 678936 x z - 144562 x - 45000 y^{2} - 196874 y z + 45000 y - 253124 z^{2} + 262124 z - 2250\right)}{375}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{281251 x^{3}}{375} - \tfrac{205313 x^{2} y}{75} - \tfrac{323814 x^{2} z}{125} + \tfrac{164588 x^{2}}{75} - \tfrac{1119377 x y^{2}}{375} - \tfrac{2404132 x y z}{375} + \tfrac{1921504 x y}{375} - \tfrac{236701 x z^{2}}{75} + \tfrac{1976632 x z}{375} - \tfrac{799877 x}{375} - \tfrac{374063 y^{3}}{375} - \tfrac{268538 y^{2} z}{75} + \tfrac{336188 y^{2}}{125} - \tfrac{1461941 y z^{2}}{375} + \tfrac{469576 y z}{75} - \tfrac{297563 y}{125} - \tfrac{164438 z^{3}}{125} + \tfrac{1238066 z^{2}}{375} - \tfrac{200588 z}{75} + \tfrac{258188}{375}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{16}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{4},\tfrac{1}{4},\tfrac{1}{4})\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle \begin{cases} 4 z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\4 y&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\4 x&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- 4 x - 4 y - 4 z + 4&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{17}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{4},\tfrac{1}{4},\tfrac{1}{4})\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 4 z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\4 y&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\4 x&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- 4 x - 4 y - 4 z + 4&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{18}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{4},\tfrac{1}{4},\tfrac{1}{4})\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 4 z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\4 y&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\4 x&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- 4 x - 4 y - 4 z + 4&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
  • \(R\) is the reference tetrahedron. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(\begin{cases} \left(\begin{array}{c}\displaystyle 2 x^{2} + 4 x y + 8 x z - 3 x + 2 y^{2} + 8 y z - 3 y + 8 z^{2} - 6 z + 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 2 x^{2} + 8 x y + 4 x z - 3 x + 8 y^{2} + 8 y z - 6 y + 2 z^{2} - 3 z + 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 8 x^{2} + 8 x y + 8 x z - 6 x + 2 y^{2} + 4 y z - 3 y + 2 z^{2} - 3 z + 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 x^{2} + 4 x y + 8 x z - 3 x + 2 y^{2} + 8 y z - 3 y + 8 z^{2} - 6 z + 1\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 x^{2} + 8 x y + 4 x z - 3 x + 8 y^{2} + 8 y z - 6 y + 2 z^{2} - 3 z + 1\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 8 x^{2} + 8 x y + 8 x z - 6 x + 2 y^{2} + 4 y z - 3 y + 2 z^{2} - 3 z + 1\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 2 x^{2} + 4 x y + 8 x z - 3 x + 2 y^{2} + 8 y z - 3 y + 8 z^{2} - 6 z + 1\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 2 x^{2} + 8 x y + 4 x z - 3 x + 8 y^{2} + 8 y z - 6 y + 2 z^{2} - 3 z + 1\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 8 x^{2} + 8 x y + 8 x z - 6 x + 2 y^{2} + 4 y z - 3 y + 2 z^{2} - 3 z + 1\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 2 x^{2} - 4 x z - x + 2 z^{2} + z\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 2 x^{2} - 4 x y - x + 2 y^{2} + y\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 8 x^{2} + 8 x y + 8 x z - 10 x + 2 y^{2} + 4 y z - 5 y + 2 z^{2} - 5 z + 3\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 x^{2} - 4 x z - x + 2 z^{2} + z\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 x^{2} - 4 x y - x + 2 y^{2} + y\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 8 x^{2} + 8 x y + 8 x z - 10 x + 2 y^{2} + 4 y z - 5 y + 2 z^{2} - 5 z + 3\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 2 x^{2} - 4 x z - x + 2 z^{2} + z\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 2 x^{2} - 4 x y - x + 2 y^{2} + y\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 8 x^{2} + 8 x y + 8 x z - 10 x + 2 y^{2} + 4 y z - 5 y + 2 z^{2} - 5 z + 3\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 2 y^{2} - 4 y z - y + 2 z^{2} + z\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 2 x^{2} - 4 x y + x + 2 y^{2} - y\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 2 x^{2} + 8 x y + 4 x z - 5 x + 8 y^{2} + 8 y z - 10 y + 2 z^{2} - 5 z + 3\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 y^{2} - 4 y z - y + 2 z^{2} + z\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 x^{2} - 4 x y + x + 2 y^{2} - y\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 x^{2} + 8 x y + 4 x z - 5 x + 8 y^{2} + 8 y z - 10 y + 2 z^{2} - 5 z + 3\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 2 y^{2} - 4 y z - y + 2 z^{2} + z\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 2 x^{2} - 4 x y + x + 2 y^{2} - y\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 2 x^{2} + 8 x y + 4 x z - 5 x + 8 y^{2} + 8 y z - 10 y + 2 z^{2} - 5 z + 3\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 2 y^{2} - 4 y z + y + 2 z^{2} - z\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 2 x^{2} - 4 x z + x + 2 z^{2} - z\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 2 x^{2} + 4 x y + 8 x z - 5 x + 2 y^{2} + 8 y z - 5 y + 8 z^{2} - 10 z + 3\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 y^{2} - 4 y z + y + 2 z^{2} - z\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 x^{2} - 4 x z + x + 2 z^{2} - z\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 x^{2} + 4 x y + 8 x z - 5 x + 2 y^{2} + 8 y z - 5 y + 8 z^{2} - 10 z + 3\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 2 y^{2} - 4 y z + y + 2 z^{2} - z\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 2 x^{2} - 4 x z + x + 2 z^{2} - z\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 2 x^{2} + 4 x y + 8 x z - 5 x + 2 y^{2} + 8 y z - 5 y + 8 z^{2} - 10 z + 3\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 4 z \left(8 z - 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 4 y \left(8 y - 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 4 x \left(8 x - 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 32 x^{2} + 64 x y + 64 x z - 60 x + 32 y^{2} + 64 y z - 60 y + 32 z^{2} - 60 z + 28\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 z \left(8 z - 1\right)\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 y \left(8 y - 1\right)\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 x \left(8 x - 1\right)\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 32 x^{2} + 64 x y + 64 x z - 60 x + 32 y^{2} + 64 y z - 60 y + 32 z^{2} - 60 z + 28\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 z \left(8 z - 1\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 y \left(8 y - 1\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 x \left(8 x - 1\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 32 x^{2} + 64 x y + 64 x z - 60 x + 32 y^{2} + 64 y z - 60 y + 32 z^{2} - 60 z + 28\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 4 x y - 4 x z - 4 y z + 4 z^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 8 x^{2} + 20 x y + 12 x z - 12 x + 8 y^{2} + 12 y z - 12 y + 4 z^{2} - 8 z + 4\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 x y - 4 x z - 4 y z + 4 z^{2}\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 8 x^{2} + 20 x y + 12 x z - 12 x + 8 y^{2} + 12 y z - 12 y + 4 z^{2} - 8 z + 4\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 x y - 4 x z - 4 y z + 4 z^{2}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 8 x^{2} + 20 x y + 12 x z - 12 x + 8 y^{2} + 12 y z - 12 y + 4 z^{2} - 8 z + 4\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle - 4 x^{2} - 4 x y - 4 x z + 4 x + 4 y z + 8 z^{2} - 4 z\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - 4 x^{2} - 4 x y - 4 x z + 4 x + 8 y^{2} + 4 y z - 4 y\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle - 4 x^{2} - 4 x y - 4 x z + 4 x + 4 y z + 8 z^{2} - 4 z\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle - 4 x^{2} - 4 x y - 4 x z + 4 x + 8 y^{2} + 4 y z - 4 y\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - 4 x^{2} - 4 x y - 4 x z + 4 x + 4 y z + 8 z^{2} - 4 z\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - 4 x^{2} - 4 x y - 4 x z + 4 x + 8 y^{2} + 4 y z - 4 y\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle - 4 x y + 4 x z - 4 y^{2} - 4 y z + 4 y + 8 z^{2} - 4 z\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 8 x^{2} - 4 x y + 4 x z - 4 x - 4 y^{2} - 4 y z + 4 y\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle - 4 x y + 4 x z - 4 y^{2} - 4 y z + 4 y + 8 z^{2} - 4 z\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 8 x^{2} - 4 x y + 4 x z - 4 x - 4 y^{2} - 4 y z + 4 y\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - 4 x y + 4 x z - 4 y^{2} - 4 y z + 4 y + 8 z^{2} - 4 z\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 8 x^{2} - 4 x y + 4 x z - 4 x - 4 y^{2} - 4 y z + 4 y\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - 4 x y + 4 x z + 4 y^{2} - 4 y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 8 x^{2} + 12 x y + 20 x z - 12 x + 4 y^{2} + 12 y z - 8 y + 8 z^{2} - 12 z + 4\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle - 4 x y + 4 x z + 4 y^{2} - 4 y z\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 8 x^{2} + 12 x y + 20 x z - 12 x + 4 y^{2} + 12 y z - 8 y + 8 z^{2} - 12 z + 4\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - 4 x y + 4 x z + 4 y^{2} - 4 y z\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 8 x^{2} + 12 x y + 20 x z - 12 x + 4 y^{2} + 12 y z - 8 y + 8 z^{2} - 12 z + 4\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 4 x y - 4 x z + 8 y^{2} - 4 y z - 4 y - 4 z^{2} + 4 z\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 8 x^{2} + 4 x y - 4 x z - 4 x - 4 y z - 4 z^{2} + 4 z\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 x y - 4 x z + 8 y^{2} - 4 y z - 4 y - 4 z^{2} + 4 z\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 8 x^{2} + 4 x y - 4 x z - 4 x - 4 y z - 4 z^{2} + 4 z\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 x y - 4 x z + 8 y^{2} - 4 y z - 4 y - 4 z^{2} + 4 z\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 8 x^{2} + 4 x y - 4 x z - 4 x - 4 y z - 4 z^{2} + 4 z\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 4 x^{2} - 4 x y - 4 x z + 4 y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 4 x^{2} + 12 x y + 12 x z - 8 x + 8 y^{2} + 20 y z - 12 y + 8 z^{2} - 12 z + 4\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 x^{2} - 4 x y - 4 x z + 4 y z\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 x^{2} + 12 x y + 12 x z - 8 x + 8 y^{2} + 20 y z - 12 y + 8 z^{2} - 12 z + 4\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 x^{2} - 4 x y - 4 x z + 4 y z\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 x^{2} + 12 x y + 12 x z - 8 x + 8 y^{2} + 20 y z - 12 y + 8 z^{2} - 12 z + 4\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 16 z \left(- x - y - 2 z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 16 y \left(- x - 2 y - z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 16 x \left(- 2 x - y - z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 16 z \left(- x - y - 2 z + 1\right)\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 16 y \left(- x - 2 y - z + 1\right)\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 16 x \left(- 2 x - y - z + 1\right)\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 16 z \left(- x - y - 2 z + 1\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 16 y \left(- x - 2 y - z + 1\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 16 x \left(- 2 x - y - z + 1\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 16 y \left(- y + z\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 16 x \left(- x + z\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - 16 x^{2} - 32 x y - 48 x z + 32 x - 16 y^{2} - 48 y z + 32 y - 32 z^{2} + 48 z - 16\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 16 y \left(- y + z\right)\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 16 x \left(- x + z\right)\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle - 16 x^{2} - 32 x y - 48 x z + 32 x - 16 y^{2} - 48 y z + 32 y - 32 z^{2} + 48 z - 16\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 16 y \left(- y + z\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 16 x \left(- x + z\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - 16 x^{2} - 32 x y - 48 x z + 32 x - 16 y^{2} - 48 y z + 32 y - 32 z^{2} + 48 z - 16\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 16 z \left(y - z\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 16 x \left(- x + y\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - 16 x^{2} - 48 x y - 32 x z + 32 x - 32 y^{2} - 48 y z + 48 y - 16 z^{2} + 32 z - 16\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 16 z \left(y - z\right)\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 16 x \left(- x + y\right)\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle - 16 x^{2} - 48 x y - 32 x z + 32 x - 32 y^{2} - 48 y z + 48 y - 16 z^{2} + 32 z - 16\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 16 z \left(y - z\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 16 x \left(- x + y\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - 16 x^{2} - 48 x y - 32 x z + 32 x - 32 y^{2} - 48 y z + 48 y - 16 z^{2} + 32 z - 16\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 16 z \left(x - z\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 16 y \left(x - y\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - 32 x^{2} - 48 x y - 48 x z + 48 x - 16 y^{2} - 32 y z + 32 y - 16 z^{2} + 32 z - 16\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 16 z \left(x - z\right)\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 16 y \left(x - y\right)\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle - 32 x^{2} - 48 x y - 48 x z + 48 x - 16 y^{2} - 32 y z + 32 y - 16 z^{2} + 32 z - 16\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 16 z \left(x - z\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 16 y \left(x - y\right)\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - 32 x^{2} - 48 x y - 48 x z + 48 x - 16 y^{2} - 32 y z + 32 y - 16 z^{2} + 32 z - 16\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle \frac{z \left(- 183750 x^{2} - 143936 x y - 222439 x z + 174750 x + 221968 y z + 428657 z^{2} - 208875 z + 2250\right)}{375}\\\displaystyle \frac{z \left(- 163564 x y + 231782 x z - 183750 y^{2} - 202811 y z + 174750 y + 418843 z^{2} - 208875 z + 2250\right)}{375}\\\displaystyle \frac{2 z \left(132766 x z + 127859 y z + 70875 z^{2} - 87375 z + 1125\right)}{375}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \frac{y \left(- 183750 x^{2} - 109061 x y - 257314 x z + 174750 x + 371968 y^{2} + 278657 y z - 208875 y + 2250\right)}{375}\\\displaystyle \frac{y \left(68218 x y + 1125 y^{2} + 31157 y z - 34125 y + 2250\right)}{375}\\\displaystyle \frac{2 y \left(17234 x y + 115532 x z + 97141 y^{2} + 52843 y z - 34125 y + 48750 z^{2} - 53250 z + 1125\right)}{375}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \frac{x \left(1125 x^{2} + 78032 x y + 21343 x z - 34125 x + 2250\right)}{375}\\\displaystyle \frac{x \left(381782 x^{2} - 128689 x y + 268843 x z - 208875 x - 183750 y^{2} - 237686 y z + 174750 y + 2250\right)}{375}\\\displaystyle \frac{2 x \left(92234 x^{2} + 22141 x y + 62657 x z - 34125 x + 105718 y z + 48750 z^{2} - 53250 z + 1125\right)}{375}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 263 x^{3} + \frac{91468 x^{2} y}{125} + \frac{148157 x^{2} z}{125} - 699 x^{2} + \frac{277747 x y^{2}}{375} + 2166 x y z - \frac{515122 x y}{375} + \frac{504503 x z^{2}}{375} - \frac{741878 x z}{375} + 627 x + \frac{101968 y^{3}}{375} + \frac{377593 y^{2} z}{375} - \frac{277811 y^{2}}{375} + \frac{434282 y z^{2}}{375} - 1824 y z + \frac{247468 y}{375} + \frac{158657 z^{3}}{375} - \frac{391189 z^{2}}{375} + \frac{304157 z}{375} - 191\\\displaystyle \frac{111782 x^{3}}{375} + \frac{317003 x^{2} y}{375} + \frac{387407 x^{2} z}{375} - \frac{297439 x^{2}}{375} + \frac{101282 x y^{2}}{125} + 2166 x y z - \frac{554378 x y}{375} + \frac{424468 x z^{2}}{375} - 1824 x z + \frac{257282 x}{375} + 263 y^{3} + \frac{138343 y^{2} z}{125} - 699 y^{2} + \frac{465247 y z^{2}}{375} - \frac{702622 y z}{375} + 627 y + \frac{148843 z^{3}}{375} - \frac{371561 z^{2}}{375} + \frac{294343 z}{375} - 191\\\displaystyle - \frac{85532 x^{3}}{375} - \frac{231782 x^{2} y}{375} - \frac{612878 x^{2} z}{375} + \frac{237814 x^{2}}{375} - \frac{221968 x y^{2}}{375} - 3084 x y z + 1176 x y - \frac{283532 x z^{2}}{125} + \frac{1078628 x z}{375} - \frac{221282 x}{375} - \frac{75718 y^{3}}{375} - \frac{573622 y^{2} z}{375} + \frac{218186 y^{2}}{375} - \frac{273718 y z^{2}}{125} + \frac{1039372 y z}{375} - \frac{211468 y}{375} - 862 z^{3} + 1926 z^{2} - 1248 z + 184\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle \frac{z \left(- 20250 x^{2} - 59434 x y - 39791 x z + 22050 x - 9000 y^{2} + 16217 y z + 9000 y + 50158 z^{2} - 13725 z - 450\right)}{75}\\\displaystyle \frac{z \left(68434 x y - 7217 x z + 38250 y^{2} + 37091 y z - 36450 y - 45658 z^{2} + 15525 z - 450\right)}{75}\\\displaystyle \frac{z \left(- 13967 x z - 8533 y z + 900 z^{2} + 7200 z - 450\right)}{75}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \frac{y \left(- 20250 x^{2} - 133909 x y + 34684 x z + 22050 x + 97217 y^{2} - 30842 y z - 13725 y - 9000 z^{2} + 9000 z - 450\right)}{75}\\\displaystyle \frac{y \left(61217 x y + 29025 y^{2} + 658 y z - 20925 y - 450\right)}{75}\\\displaystyle \frac{y \left(67967 x y - 81934 x z - 36467 y^{2} + 46834 y z - 12600 y - 18000 z^{2} + 19800 z - 450\right)}{75}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - 303 x^{3} - \frac{34217 x^{2} y}{75} + \frac{12842 x^{2} z}{75} + 111 x^{2} - 6 x - 120 y^{2} z - 120 y z^{2} + 120 y z\\\displaystyle \frac{x \left(- 43217 x^{2} + 95209 x y - 9658 x z + 15525 x + 38250 y^{2} + 10316 y z - 36450 y - 450\right)}{75}\\\displaystyle \frac{x \left(31967 x^{2} - 467 x y - 27034 x z - 12600 x - 8066 y z - 18000 z^{2} + 19800 z - 450\right)}{75}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - 1923 x^{3} - \frac{107683 x^{2} y}{25} - \frac{154742 x^{2} z}{25} + 4719 x^{2} - \frac{243607 x y^{2}}{75} - 9246 x y z + \frac{520582 x y}{75} - \frac{431843 x z^{2}}{75} + \frac{708818 x z}{75} - 3687 x - \frac{64783 y^{3}}{75} - \frac{259408 y^{2} z}{75} + \frac{196841 y^{2}}{75} - \frac{306467 y z^{2}}{75} + 6744 y z - \frac{198883 y}{75} - \frac{111842 z^{3}}{75} + \frac{290959 z^{2}}{75} - \frac{245942 z}{75} + 891\\\displaystyle \frac{28783 x^{3}}{75} + \frac{51907 x^{2} y}{75} + \frac{83908 x^{2} z}{75} - \frac{78041 x^{2}}{75} + \frac{15883 x y^{2}}{25} + 2094 x y z - \frac{119182 x y}{75} + \frac{81467 x z^{2}}{75} - 2016 x z + \frac{70183 x}{75} + 327 y^{3} + \frac{34442 y^{2} z}{25} - 951 y^{2} + \frac{105143 y z^{2}}{75} - \frac{172418 y z}{75} + 903 y + \frac{26342 z^{3}}{75} - \frac{73159 z^{2}}{75} + \frac{67742 z}{75} - 279\\\displaystyle \frac{103967 x^{3}}{75} + \frac{243467 x^{2} y}{75} + \frac{380768 x^{2} z}{75} - \frac{256534 x^{2}}{75} + \frac{175033 x y^{2}}{75} + 7344 x y z - 5016 x y + \frac{128567 x z^{2}}{25} - \frac{588668 x z}{75} + \frac{201617 x}{75} + \frac{35533 y^{3}}{75} + \frac{170032 y^{2} z}{75} - \frac{119666 y^{2}}{75} + \frac{81133 y z^{2}}{25} - \frac{377932 y z}{75} + \frac{133183 y}{75} + 1452 z^{3} - 3576 z^{2} + 2778 z - 654\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 6 z \left(40 x^{2} + 72 x y + 6 x z - 44 x - 96 y z + 26 z^{2} + 28 z + 1\right)\\\displaystyle \frac{2 z \left(22500 x^{2} - 24262 x y + 45881 x z - 22500 x - 10762 y^{2} + 118857 y z + 6262 y - 119738 z^{2} + 3619 z + 1125\right)}{375}\\\displaystyle \frac{2 z \left(- 32869 x z - 29738 y z - 41869 z^{2} + 21619 z + 1125\right)}{375}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 6 y \left(40 x^{2} - 50 x y + 128 x z - 44 x - 16 y^{2} - 54 y z + 28 y + 1\right)\\\displaystyle 120 x^{2} z - \frac{1762 x y^{2}}{375} + 120 x z^{2} - 120 x z + \frac{52238 y^{3}}{375} - \frac{75524 y^{2} z}{375} + \frac{19762 y^{2}}{375} + 6 y\\\displaystyle \frac{2 y \left(55369 x y - 88238 x z - 15262 y^{2} - 22107 y z - 8119 y - 34238 z^{2} + 29738 z + 1125\right)}{375}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 6 x \left(- 2 x^{2} - 24 x y + 74 x z - 16 x + 1\right)\\\displaystyle \frac{2 x \left(- 156619 x^{2} + 192619 x y + 82762 x z + 3619 x - 10762 y^{2} - 98024 y z + 6262 y + 22500 z^{2} - 22500 z + 1125\right)}{375}\\\displaystyle \frac{2 x \left(145369 x^{2} - 105262 x y - 185869 x z - 8119 x + 75524 y z - 34238 z^{2} + 29738 z + 1125\right)}{375}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - 1452 x^{3} - 4152 x^{2} y - 4236 x^{2} z + 3576 x^{2} - 3756 x y^{2} - 7344 x y z + 6528 x y - 3588 x z^{2} + 6360 x z - 2778 x - 1056 y^{3} - 2916 y^{2} z + 2760 y^{2} - 2664 y z^{2} + 5016 y z - 2358 y - 804 z^{3} + 2256 z^{2} - 2106 z + 654\\\displaystyle \frac{496762 x^{3}}{375} + \frac{394162 x^{2} y}{75} + \frac{1654048 x^{2} z}{375} - \frac{463762 x^{2}}{125} + \frac{2379334 x y^{2}}{375} + \frac{4326668 x y z}{375} - \frac{732524 x y}{75} + \frac{345562 x z^{2}}{75} - \frac{3020096 x z}{375} + \frac{430012 x}{125} + \frac{301762 y^{3}}{125} + \frac{520124 y^{2} z}{75} - \frac{2199334 y^{2}}{375} + \frac{755286 y z^{2}}{125} - \frac{3957668 y z}{375} + \frac{337912 y}{75} + \frac{570524 z^{3}}{375} - \frac{307762 z^{2}}{75} + \frac{1363798 z}{375} - \frac{395512}{375}\\\displaystyle - \frac{69262 x^{3}}{375} - \frac{529048 x^{2} y}{375} - \frac{67462 x^{2} z}{75} + \frac{100762 x^{2}}{125} - \frac{170062 x y^{2}}{75} - \frac{1644668 x y z}{375} + \frac{1247096 x y}{375} - \frac{574834 x z^{2}}{375} + \frac{196124 x z}{75} - \frac{133012 x}{125} - \frac{390524 y^{3}}{375} - \frac{435786 y^{2} z}{125} + \frac{188962 y^{2}}{75} - \frac{244724 y z^{2}}{75} + \frac{1950668 y z}{375} - \frac{720298 y}{375} - \frac{102262 z^{3}}{125} + \frac{786334 z^{2}}{375} - \frac{129112 z}{75} + \frac{166012}{375}\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle \frac{z \left(343124 x^{2} + 421874 x y + 485436 x z - 334124 x - 233437 y z - 258749 z^{2} + 153562 z - 2250\right)}{375}\\\displaystyle 3 z \left(- 230 x y + 95 x z - 80 y^{2} + 74 y z + 88 y + 45 z^{2} - 56 z - 2\right)\\\displaystyle - 120 x^{2} y - 120 x y^{2} + 120 x y - \frac{213749 x z^{2}}{375} - \frac{120937 y z^{2}}{375} - \frac{189562 z^{3}}{375} + \frac{117562 z^{2}}{375} - 6 z\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \frac{y \left(343124 x^{2} + 479812 x y + 427498 x z - 334124 x - 413437 y^{2} - 78749 y z + 153562 y - 2250\right)}{375}\\\displaystyle 3 y \left(- 135 x y + 4 y^{2} + 35 y z + 32 y - 2\right)\\\displaystyle \frac{y \left(- 45000 x^{2} + 123749 x y - 382498 x z + 45000 x + 435937 y^{2} - 493312 y z - 144562 y - 253124 z^{2} + 262124 z - 2250\right)}{375}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \frac{x \left(221062 x^{2} + 188437 x y + 348749 x z - 180562 x - 2250\right)}{375}\\\displaystyle 3 x \left(- 65 x^{2} + 14 x y + 205 x z - 56 x - 80 y^{2} - 170 y z + 88 y - 2\right)\\\displaystyle \frac{x \left(528749 x^{2} + 30937 x y - 678936 x z - 144562 x - 45000 y^{2} - 196874 y z + 45000 y - 253124 z^{2} + 262124 z - 2250\right)}{375}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - \frac{35062 x^{3}}{125} - \frac{342559 x^{2} y}{375} - \frac{38699 x^{2} z}{75} + \frac{174934 x^{2}}{375} - \frac{58162 x y^{2}}{75} - \frac{277868 x y z}{375} + \frac{66824 x y}{75} + \frac{4314 x z^{2}}{125} + \frac{30368 x z}{375} - \frac{8212 x}{75} - \frac{53437 y^{3}}{375} - \frac{5623 y^{2} z}{375} + \frac{26812 y^{2}}{125} + \frac{29813 y z^{2}}{75} - \frac{148504 y z}{375} + \frac{563 y}{125} + \frac{101251 z^{3}}{375} - \frac{45788 z^{2}}{75} + \frac{156377 z}{375} - \frac{28688}{375}\\\displaystyle 765 x^{3} + 3432 x^{2} y + 2625 x^{2} z - 2178 x^{2} + 4119 x y^{2} + 7344 x y z - 6204 x y + 2955 x z^{2} - 5016 x z + 2067 x + 1452 y^{3} + 4269 y^{2} z - 3576 y^{2} + 3912 y z^{2} - 6684 y z + 2778 y + 1095 z^{3} - 2838 z^{2} + 2397 z - 654\\\displaystyle - \frac{281251 x^{3}}{375} - \frac{205313 x^{2} y}{75} - \frac{323814 x^{2} z}{125} + \frac{164588 x^{2}}{75} - \frac{1119377 x y^{2}}{375} - \frac{2404132 x y z}{375} + \frac{1921504 x y}{375} - \frac{236701 x z^{2}}{75} + \frac{1976632 x z}{375} - \frac{799877 x}{375} - \frac{374063 y^{3}}{375} - \frac{268538 y^{2} z}{75} + \frac{336188 y^{2}}{125} - \frac{1461941 y z^{2}}{375} + \frac{469576 y z}{75} - \frac{297563 y}{125} - \frac{164438 z^{3}}{125} + \frac{1238066 z^{2}}{375} - \frac{200588 z}{75} + \frac{258188}{375}\end{array}\right)&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)
  • \(\mathcal{L}=\{l_0,...,l_{48}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,0)\cdot\left(\begin{array}{c}1\\1\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle \begin{cases} 2 x^{2} - 4 x z - x + 2 z^{2} + z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\2 x^{2} - 4 x y - x + 2 y^{2} + y&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\8 x^{2} + 8 x y + 8 x z - 10 x + 2 y^{2} + 4 y z - 5 y + 2 z^{2} - 5 z + 3&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,0)\cdot\left(\begin{array}{c}1\\1\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 2 y^{2} - 4 y z - y + 2 z^{2} + z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\2 x^{2} - 4 x y + x + 2 y^{2} - y&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\2 x^{2} + 8 x y + 4 x z - 5 x + 8 y^{2} + 8 y z - 10 y + 2 z^{2} - 5 z + 3&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,1)\cdot\left(\begin{array}{c}1\\1\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\2 y^{2} - 4 y z + y + 2 z^{2} - z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\2 x^{2} - 4 x z + x + 2 z^{2} - z&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\2 x^{2} + 4 x y + 8 x z - 5 x + 2 y^{2} + 8 y z - 5 y + 8 z^{2} - 10 z + 3&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle \begin{cases} 2 x^{2} + 4 x y + 8 x z - 3 x + 2 y^{2} + 8 y z - 3 y + 8 z^{2} - 6 z + 1&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\2 x^{2} + 8 x y + 4 x z - 3 x + 8 y^{2} + 8 y z - 6 y + 2 z^{2} - 3 z + 1&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\8 x^{2} + 8 x y + 8 x z - 6 x + 2 y^{2} + 4 y z - 3 y + 2 z^{2} - 3 z + 1&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle \begin{cases} 2 y^{2} - 4 y z - y + 2 z^{2} + z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\2 x^{2} - 4 x y + x + 2 y^{2} - y&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\2 x^{2} + 8 x y + 4 x z - 5 x + 8 y^{2} + 8 y z - 10 y + 2 z^{2} - 5 z + 3&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} - 2 y^{2} + 4 y z + y - 2 z^{2} - z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- 2 x^{2} + 4 x y - x - 2 y^{2} + y&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- 2 x^{2} - 8 x y - 4 x z + 5 x - 8 y^{2} - 8 y z + 10 y - 2 z^{2} + 5 z - 3&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,1)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\2 y^{2} - 4 y z + y + 2 z^{2} - z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\2 x^{2} - 4 x z + x + 2 z^{2} - z&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\2 x^{2} + 4 x y + 8 x z - 5 x + 2 y^{2} + 8 y z - 5 y + 8 z^{2} - 10 z + 3&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\- 2 y^{2} + 4 y z - y - 2 z^{2} + z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- 2 x^{2} + 4 x z - x - 2 z^{2} + z&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- 2 x^{2} - 4 x y - 8 x z + 5 x - 2 y^{2} - 8 y z + 5 y - 8 z^{2} + 10 z - 3&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,0)\cdot\left(\begin{array}{c}0\\-1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} - 2 x^{2} - 4 x y - 8 x z + 3 x - 2 y^{2} - 8 y z + 3 y - 8 z^{2} + 6 z - 1&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\- 2 x^{2} - 8 x y - 4 x z + 3 x - 8 y^{2} - 8 y z + 6 y - 2 z^{2} + 3 z - 1&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- 8 x^{2} - 8 x y - 8 x z + 6 x - 2 y^{2} - 4 y z + 3 y - 2 z^{2} + 3 z - 1&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,0)\cdot\left(\begin{array}{c}0\\-1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle \begin{cases} 2 x^{2} - 4 x z - x + 2 z^{2} + z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\2 x^{2} - 4 x y - x + 2 y^{2} + y&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\8 x^{2} + 8 x y + 8 x z - 10 x + 2 y^{2} + 4 y z - 5 y + 2 z^{2} - 5 z + 3&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} - 2 x^{2} + 4 x z + x - 2 z^{2} - z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\- 2 x^{2} + 4 x y + x - 2 y^{2} - y&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- 8 x^{2} - 8 x y - 8 x z + 10 x - 2 y^{2} - 4 y z + 5 y - 2 z^{2} + 5 z - 3&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,1)\cdot\left(\begin{array}{c}0\\-1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\- 2 y^{2} + 4 y z - y - 2 z^{2} + z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- 2 x^{2} + 4 x z - x - 2 z^{2} + z&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- 2 x^{2} - 4 x y - 8 x z + 5 x - 2 y^{2} - 8 y z + 5 y - 8 z^{2} + 10 z - 3&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\2 y^{2} - 4 y z + y + 2 z^{2} - z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\2 x^{2} - 4 x z + x + 2 z^{2} - z&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\2 x^{2} + 4 x y + 8 x z - 5 x + 2 y^{2} + 8 y z - 5 y + 8 z^{2} - 10 z + 3&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{9}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 2 x^{2} + 4 x y + 8 x z - 3 x + 2 y^{2} + 8 y z - 3 y + 8 z^{2} - 6 z + 1&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\2 x^{2} + 8 x y + 4 x z - 3 x + 8 y^{2} + 8 y z - 6 y + 2 z^{2} - 3 z + 1&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\8 x^{2} + 8 x y + 8 x z - 6 x + 2 y^{2} + 4 y z - 3 y + 2 z^{2} - 3 z + 1&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{10}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle \begin{cases} - 2 x^{2} + 4 x z + x - 2 z^{2} - z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\- 2 x^{2} + 4 x y + x - 2 y^{2} - y&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- 8 x^{2} - 8 x y - 8 x z + 10 x - 2 y^{2} - 4 y z + 5 y - 2 z^{2} + 5 z - 3&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 2 x^{2} - 4 x z - x + 2 z^{2} + z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\2 x^{2} - 4 x y - x + 2 y^{2} + y&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\8 x^{2} + 8 x y + 8 x z - 10 x + 2 y^{2} + 4 y z - 5 y + 2 z^{2} - 5 z + 3&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{11}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} - 2 y^{2} + 4 y z + y - 2 z^{2} - z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- 2 x^{2} + 4 x y - x - 2 y^{2} + y&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- 2 x^{2} - 8 x y - 4 x z + 5 x - 8 y^{2} - 8 y z + 10 y - 2 z^{2} + 5 z - 3&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 2 y^{2} - 4 y z - y + 2 z^{2} + z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\2 x^{2} - 4 x y + x + 2 y^{2} - y&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\2 x^{2} + 8 x y + 4 x z - 5 x + 8 y^{2} + 8 y z - 10 y + 2 z^{2} - 5 z + 3&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{12}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{2},\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\-1\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- 2 x^{2} + 2 x y + 2 x z - 2 y z&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- 2 x^{2} - 6 x y - 6 x z + 4 x - 4 y^{2} - 10 y z + 6 y - 4 z^{2} + 6 z - 2&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\2 x^{2} - 2 x y - 2 x z + 2 y z&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\2 x^{2} + 6 x y + 6 x z - 4 x + 4 y^{2} + 10 y z - 6 y + 4 z^{2} - 6 z + 2&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{13}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},0,\tfrac{1}{2})\cdot\left(\begin{array}{c}-1\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\2 x y - 2 x z - 2 y^{2} + 2 y z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- 4 x^{2} - 6 x y - 10 x z + 6 x - 2 y^{2} - 6 y z + 4 y - 4 z^{2} + 6 z - 2&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\- 2 x y + 2 x z + 2 y^{2} - 2 y z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\4 x^{2} + 6 x y + 10 x z - 6 x + 2 y^{2} + 6 y z - 4 y + 4 z^{2} - 6 z + 2&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{14}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},\tfrac{1}{2},0)\cdot\left(\begin{array}{c}-1\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle \begin{cases} - 2 x y + 2 x z + 2 y z - 2 z^{2}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- 4 x^{2} - 10 x y - 6 x z + 6 x - 4 y^{2} - 6 y z + 6 y - 2 z^{2} + 4 z - 2&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 2 x y - 2 x z - 2 y z + 2 z^{2}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\4 x^{2} + 10 x y + 6 x z - 6 x + 4 y^{2} + 6 y z - 6 y + 2 z^{2} - 4 z + 2&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{15}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\4 x y - 4 x z + 8 y^{2} - 4 y z - 4 y - 4 z^{2} + 4 z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\8 x^{2} + 4 x y - 4 x z - 4 x - 4 y z - 4 z^{2} + 4 z&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{16}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{2},0)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} - 4 x y + 4 x z - 4 y^{2} - 4 y z + 4 y + 8 z^{2} - 4 z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\8 x^{2} - 4 x y + 4 x z - 4 x - 4 y^{2} - 4 y z + 4 y&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with edge 4 of the reference element.
\(\displaystyle l_{17}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},0,0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle \begin{cases} - 4 x^{2} - 4 x y - 4 x z + 4 x + 4 y z + 8 z^{2} - 4 z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\- 4 x^{2} - 4 x y - 4 x z + 4 x + 8 y^{2} + 4 y z - 4 y&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with edge 5 of the reference element.
\(\displaystyle l_{18}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{2},\tfrac{1}{2})\cdot\left(\begin{array}{c}1\\1\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle \begin{cases} \tfrac{z \left(1470000 x^{2} + 1151488 x y + 1779512 x z - 1468875 x - 1775744 y z + 78032 y - 3429256 z^{2} + 1692343 z - 25125\right)}{18000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(1470000 x^{2} + 872488 x y + 2058512 x z - 1468875 x - 2975744 y^{2} - 2229256 y z + 1749032 y + 21343 z - 25125\right)}{18000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(- 9000 x^{2} - 624256 x y - 170744 x z + 202125 x + 78032 y + 21343 z - 25125\right)}{18000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{263 x^{3}}{6} - \tfrac{45734 x^{2} y}{375} - \tfrac{148157 x^{2} z}{750} + \tfrac{1927 x^{2}}{16} - \tfrac{277747 x y^{2}}{2250} - 361 x y z + \tfrac{457091 x y}{2000} - \tfrac{504503 x z^{2}}{2250} + \tfrac{498713 x z}{1500} - \tfrac{2593 x}{24} - \tfrac{50984 y^{3}}{1125} - \tfrac{377593 y^{2} z}{2250} + \tfrac{268057 y^{2}}{2250} - \tfrac{217141 y z^{2}}{1125} + \tfrac{14327 y z}{48} - \tfrac{631529 y}{6000} - \tfrac{158657 z^{3}}{2250} + \tfrac{3108169 z^{2}}{18000} - \tfrac{200399 z}{1500} + \tfrac{503}{16}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(1308512 x y - 1854256 x z + 68218 x + 1470000 y^{2} + 1622488 y z - 1468875 y - 3350744 z^{2} + 1702157 z - 25125\right)}{18000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 545744 x y + 68218 x - 9000 y^{2} - 249256 y z + 202125 y + 31157 z - 25125\right)}{18000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{190891 x^{3}}{1125} + \tfrac{128689 x^{2} y}{2250} - \tfrac{268843 x^{2} z}{2250} + \tfrac{887609 x^{2}}{9000} + \tfrac{245 x y^{2}}{3} + \tfrac{118843 x y z}{1125} - \tfrac{4013 x y}{48} - \tfrac{4843 x z}{18000} - \tfrac{67 x}{48} + 2 y z&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{55891 x^{3}}{1125} - \tfrac{317003 x^{2} y}{2250} - \tfrac{387407 x^{2} z}{2250} + \tfrac{1173647 x^{2}}{9000} - \tfrac{50641 x y^{2}}{375} - 361 x y z + \tfrac{1515227 x y}{6000} - \tfrac{212234 x z^{2}}{1125} + \tfrac{14615 x z}{48} - \tfrac{684971 x}{6000} - \tfrac{263 y^{3}}{6} - \tfrac{138343 y^{2} z}{750} + \tfrac{1991 y^{2}}{16} - \tfrac{465247 y z^{2}}{2250} + \tfrac{324483 y z}{1000} - \tfrac{2737 y}{24} - \tfrac{148843 z^{3}}{2250} + \tfrac{3013331 z^{2}}{18000} - \tfrac{404077 z}{3000} + \tfrac{535}{16}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(- 1062128 x z + 132766 x - 1022872 y z + 127859 y - 567000 z^{2} + 733875 z - 82875\right)}{9000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 137872 x y - 924256 x z + 132766 x - 777128 y^{2} - 422744 y z + 400859 y - 390000 z^{2} + 460875 z - 82875\right)}{9000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{92234 x^{3}}{1125} - \tfrac{22141 x^{2} y}{1125} - \tfrac{62657 x^{2} z}{1125} + \tfrac{211883 x^{2}}{4500} - \tfrac{105718 x y z}{1125} + \tfrac{109859 x y}{9000} - \tfrac{130 x z^{2}}{3} + \tfrac{1181 x z}{24} - \tfrac{221 x}{24} + 2 y z&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{42766 x^{3}}{1125} + \tfrac{115891 x^{2} y}{1125} + \tfrac{306439 x^{2} z}{1125} - \tfrac{533011 x^{2}}{4500} + \tfrac{110984 x y^{2}}{1125} + 514 x y z - \tfrac{5255 x y}{24} + \tfrac{141766 x z^{2}}{375} - \tfrac{492017 x z}{1000} + \tfrac{351923 x}{3000} + \tfrac{37859 y^{3}}{1125} + \tfrac{286811 y^{2} z}{1125} - \tfrac{964603 y^{2}}{9000} + \tfrac{136859 y z^{2}}{375} - \tfrac{705037 y z}{1500} + \tfrac{165601 y}{1500} + \tfrac{431 z^{3}}{3} - \tfrac{2567 z^{2}}{8} + \tfrac{2569 z}{12} - \tfrac{295}{8}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{19}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},0,\tfrac{1}{2})\cdot\left(\begin{array}{c}1\\1\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle \begin{cases} \tfrac{z \left(1470000 x^{2} + 1151488 x y + 1779512 x z - 1468875 x - 1775744 y z + 78032 y - 3429256 z^{2} + 1692343 z - 25125\right)}{18000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{245 x^{2} y}{3} + \tfrac{109061 x y^{2}}{2250} + \tfrac{128657 x y z}{1125} - \tfrac{4013 x y}{48} + 2 x z - \tfrac{185984 y^{3}}{1125} - \tfrac{278657 y^{2} z}{2250} + \tfrac{223129 y^{2}}{2250} - \tfrac{14657 y z}{18000} - \tfrac{67 y}{48}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(- 9000 x^{2} - 624256 x y - 170744 x z + 202125 x + 78032 y + 21343 z - 25125\right)}{18000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{263 x^{3}}{6} - \tfrac{45734 x^{2} y}{375} - \tfrac{148157 x^{2} z}{750} + \tfrac{1991 x^{2}}{16} - \tfrac{277747 x y^{2}}{2250} - 361 x y z + \tfrac{469091 x y}{2000} - \tfrac{504503 x z^{2}}{2250} + \tfrac{513713 x z}{1500} - \tfrac{2737 x}{24} - \tfrac{50984 y^{3}}{1125} - \tfrac{377593 y^{2} z}{2250} + \tfrac{272557 y^{2}}{2250} - \tfrac{217141 y z^{2}}{1125} + \tfrac{14615 y z}{48} - \tfrac{655529 y}{6000} - \tfrac{158657 z^{3}}{2250} + \tfrac{3180169 z^{2}}{18000} - \tfrac{209399 z}{1500} + \tfrac{535}{16}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(1308512 x y - 1854256 x z + 68218 x + 1470000 y^{2} + 1622488 y z - 1468875 y - 3350744 z^{2} + 1702157 z - 25125\right)}{18000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 545744 x y + 68218 x - 9000 y^{2} - 249256 y z + 202125 y + 31157 z - 25125\right)}{18000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(- 3054256 x^{2} + 1029512 x y - 2150744 x z + 1739218 x + 1470000 y^{2} + 1901488 y z - 1468875 y + 31157 z - 25125\right)}{18000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{55891 x^{3}}{1125} - \tfrac{317003 x^{2} y}{2250} - \tfrac{387407 x^{2} z}{2250} + \tfrac{1155647 x^{2}}{9000} - \tfrac{50641 x y^{2}}{375} - 361 x y z + \tfrac{1479227 x y}{6000} - \tfrac{212234 x z^{2}}{1125} + \tfrac{14327 x z}{48} - \tfrac{660971 x}{6000} - \tfrac{263 y^{3}}{6} - \tfrac{138343 y^{2} z}{750} + \tfrac{1927 y^{2}}{16} - \tfrac{465247 y z^{2}}{2250} + \tfrac{314483 y z}{1000} - \tfrac{2593 y}{24} - \tfrac{148843 z^{3}}{2250} + \tfrac{2941331 z^{2}}{18000} - \tfrac{386077 z}{3000} + \tfrac{503}{16}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(- 1062128 x z + 132766 x - 1022872 y z + 127859 y - 567000 z^{2} + 733875 z - 82875\right)}{9000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\- \tfrac{17234 x y^{2}}{1125} - \tfrac{115532 x y z}{1125} + \tfrac{57383 x y}{4500} + 2 x z - \tfrac{97141 y^{3}}{1125} - \tfrac{52843 y^{2} z}{1125} + \tfrac{418859 y^{2}}{9000} - \tfrac{130 y z^{2}}{3} + \tfrac{1181 y z}{24} - \tfrac{221 y}{24}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(- 737872 x^{2} - 177128 x y - 501256 x z + 405766 x - 845744 y z + 127859 y - 390000 z^{2} + 460875 z - 82875\right)}{9000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{42766 x^{3}}{1125} + \tfrac{115891 x^{2} y}{1125} + \tfrac{306439 x^{2} z}{1125} - \tfrac{524011 x^{2}}{4500} + \tfrac{110984 x y^{2}}{1125} + 514 x y z - \tfrac{5255 x y}{24} + \tfrac{141766 x z^{2}}{375} - \tfrac{488017 x z}{1000} + \tfrac{345923 x}{3000} + \tfrac{37859 y^{3}}{1125} + \tfrac{286811 y^{2} z}{1125} - \tfrac{982603 y^{2}}{9000} + \tfrac{136859 y z^{2}}{375} - \tfrac{711037 y z}{1500} + \tfrac{168601 y}{1500} + \tfrac{431 z^{3}}{3} - \tfrac{2567 z^{2}}{8} + \tfrac{2569 z}{12} - \tfrac{295}{8}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{20}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},\tfrac{1}{2},0)\cdot\left(\begin{array}{c}1\\1\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{20} = \left(\begin{array}{c}\displaystyle \begin{cases} \tfrac{245 x^{2} z}{3} + \tfrac{71968 x y z}{1125} + 2 x y + \tfrac{222439 x z^{2}}{2250} - \tfrac{4013 x z}{48} - \tfrac{110984 y z^{2}}{1125} + \tfrac{2627 y z}{1125} - \tfrac{428657 z^{3}}{2250} + \tfrac{1728343 z^{2}}{18000} - \tfrac{67 z}{48}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(1470000 x^{2} + 872488 x y + 2058512 x z - 1468875 x - 2975744 y^{2} - 2229256 y z + 1749032 y + 21343 z - 25125\right)}{18000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(- 9000 x^{2} - 624256 x y - 170744 x z + 202125 x + 78032 y + 21343 z - 25125\right)}{18000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{263 x^{3}}{6} - \tfrac{45734 x^{2} y}{375} - \tfrac{148157 x^{2} z}{750} + \tfrac{1991 x^{2}}{16} - \tfrac{277747 x y^{2}}{2250} - 361 x y z + \tfrac{477091 x y}{2000} - \tfrac{504503 x z^{2}}{2250} + \tfrac{507713 x z}{1500} - \tfrac{2737 x}{24} - \tfrac{50984 y^{3}}{1125} - \tfrac{377593 y^{2} z}{2250} + \tfrac{277057 y^{2}}{2250} - \tfrac{217141 y z^{2}}{1125} + \tfrac{14615 y z}{48} - \tfrac{667529 y}{6000} - \tfrac{158657 z^{3}}{2250} + \tfrac{3144169 z^{2}}{18000} - \tfrac{206399 z}{1500} + \tfrac{535}{16}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{81782 x y z}{1125} + 2 x y - \tfrac{115891 x z^{2}}{1125} + \tfrac{16109 x z}{9000} + \tfrac{245 y^{2} z}{3} + \tfrac{202811 y z^{2}}{2250} - \tfrac{4013 y z}{48} - \tfrac{418843 z^{3}}{2250} + \tfrac{1738157 z^{2}}{18000} - \tfrac{67 z}{48}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 545744 x y + 68218 x - 9000 y^{2} - 249256 y z + 202125 y + 31157 z - 25125\right)}{18000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(- 3054256 x^{2} + 1029512 x y - 2150744 x z + 1739218 x + 1470000 y^{2} + 1901488 y z - 1468875 y + 31157 z - 25125\right)}{18000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{55891 x^{3}}{1125} - \tfrac{317003 x^{2} y}{2250} - \tfrac{387407 x^{2} z}{2250} + \tfrac{1191647 x^{2}}{9000} - \tfrac{50641 x y^{2}}{375} - 361 x y z + \tfrac{1539227 x y}{6000} - \tfrac{212234 x z^{2}}{1125} + \tfrac{14615 x z}{48} - \tfrac{696971 x}{6000} - \tfrac{263 y^{3}}{6} - \tfrac{138343 y^{2} z}{750} + \tfrac{1991 y^{2}}{16} - \tfrac{465247 y z^{2}}{2250} + \tfrac{320483 y z}{1000} - \tfrac{2737 y}{24} - \tfrac{148843 z^{3}}{2250} + \tfrac{2977331 z^{2}}{18000} - \tfrac{398077 z}{3000} + \tfrac{535}{16}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(- 1062128 x z + 132766 x - 1022872 y z + 127859 y - 567000 z^{2} + 733875 z - 82875\right)}{9000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 137872 x y - 924256 x z + 132766 x - 777128 y^{2} - 422744 y z + 400859 y - 390000 z^{2} + 460875 z - 82875\right)}{9000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(- 737872 x^{2} - 177128 x y - 501256 x z + 405766 x - 845744 y z + 127859 y - 390000 z^{2} + 460875 z - 82875\right)}{9000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{42766 x^{3}}{1125} + \tfrac{115891 x^{2} y}{1125} + \tfrac{306439 x^{2} z}{1125} - \tfrac{542011 x^{2}}{4500} + \tfrac{110984 x y^{2}}{1125} + 514 x y z - \tfrac{5399 x y}{24} + \tfrac{141766 x z^{2}}{375} - \tfrac{498017 x z}{1000} + \tfrac{363923 x}{3000} + \tfrac{37859 y^{3}}{1125} + \tfrac{286811 y^{2} z}{1125} - \tfrac{1000603 y^{2}}{9000} + \tfrac{136859 y z^{2}}{375} - \tfrac{720037 y z}{1500} + \tfrac{174601 y}{1500} + \tfrac{431 z^{3}}{3} - \tfrac{2599 z^{2}}{8} + \tfrac{2641 z}{12} - \tfrac{311}{8}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{21}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{2},\tfrac{1}{2})\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{21} = \left(\begin{array}{c}\displaystyle \begin{cases} \tfrac{z \left(162000 x^{2} + 475472 x y + 318328 x z - 220725 x + 72000 y^{2} - 129736 y z - 97217 y - 401264 z^{2} + 131642 z + 15525\right)}{3600}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(162000 x^{2} + 1071272 x y - 277472 x z - 220725 x - 777736 y^{2} + 246736 y z + 84583 y + 72000 z^{2} - 50158 z + 15525\right)}{3600}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{101 x^{3}}{2} + \tfrac{34217 x^{2} y}{450} - \tfrac{6421 x^{2} z}{225} - \tfrac{429 x^{2}}{16} - \tfrac{39617 x y}{3600} + \tfrac{3721 x z}{1800} + \tfrac{69 x}{16} + 20 y^{2} z + 20 y z^{2} - 16 y z&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{641 x^{3}}{2} + \tfrac{107683 x^{2} y}{150} + \tfrac{77371 x^{2} z}{75} - \tfrac{12323 x^{2}}{16} + \tfrac{243607 x y^{2}}{450} + 1541 x y z - \tfrac{675319 x y}{600} + \tfrac{431843 x z^{2}}{450} - \tfrac{1868287 x z}{1200} + \tfrac{4727 x}{8} + \tfrac{64783 y^{3}}{450} + \tfrac{129704 y^{2} z}{225} - \tfrac{1520711 y^{2}}{3600} + \tfrac{306467 y z^{2}}{450} - \tfrac{17649 y z}{16} + \tfrac{83929 y}{200} + \tfrac{55921 z^{3}}{225} - \tfrac{1160357 z^{2}}{1800} + \tfrac{214917 z}{400} - \tfrac{2259}{16}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(- 547472 x y + 57736 x z + 61217 x - 306000 y^{2} - 296728 y z + 335025 y + 365264 z^{2} - 123542 z - 22725\right)}{3600}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 489736 x y + 61217 x - 232200 y^{2} - 5264 y z + 210825 y + 658 z - 22725\right)}{3600}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{43217 x^{3}}{450} - \tfrac{95209 x^{2} y}{450} + \tfrac{4829 x^{2} z}{225} - \tfrac{70183 x^{2}}{3600} - 85 x y^{2} - \tfrac{5158 x y z}{225} + \tfrac{1521 x y}{16} + \tfrac{3929 x z}{1800} - \tfrac{101 x}{16} - 2 y z&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{28783 x^{3}}{450} - \tfrac{51907 x^{2} y}{450} - \tfrac{41954 x^{2} z}{225} + \tfrac{555911 x^{2}}{3600} - \tfrac{15883 x y^{2}}{150} - 349 x y z + \tfrac{137869 x y}{600} - \tfrac{81467 x z^{2}}{450} + \tfrac{5005 x z}{16} - \tfrac{25529 x}{200} - \tfrac{109 y^{3}}{2} - \tfrac{17221 y^{2} z}{75} + \tfrac{2279 y^{2}}{16} - \tfrac{105143 y z^{2}}{450} + \tfrac{433087 y z}{1200} - \tfrac{1001 y}{8} - \tfrac{13171 z^{3}}{225} + \tfrac{285107 z^{2}}{1800} - \tfrac{54817 z}{400} + \tfrac{595}{16}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(111736 x z - 13967 x + 68264 y z - 8533 y - 7200 z^{2} - 42300 z + 5400\right)}{3600}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 543736 x y + 655472 x z - 13967 x + 291736 y^{2} - 374672 y z + 92267 y + 144000 z^{2} - 143100 z + 5400\right)}{3600}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{31967 x^{3}}{450} + \tfrac{467 x^{2} y}{450} + \tfrac{13517 x^{2} z}{225} + \tfrac{79633 x^{2}}{3600} + \tfrac{4033 x y z}{225} - \tfrac{1333 x y}{3600} + 40 x z^{2} - \tfrac{151 x z}{4} + \tfrac{3 x}{2} - 2 y z&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{103967 x^{3}}{450} - \tfrac{243467 x^{2} y}{450} - \tfrac{190384 x^{2} z}{225} + \tfrac{2059039 x^{2}}{3600} - \tfrac{175033 x y^{2}}{450} - 1224 x y z + \tfrac{3345 x y}{4} - \tfrac{128567 x z^{2}}{150} + \tfrac{1562137 x z}{1200} - \tfrac{179367 x}{400} - \tfrac{35533 y^{3}}{450} - \tfrac{85016 y^{2} z}{225} + \tfrac{951461 y^{2}}{3600} - \tfrac{81133 y z^{2}}{150} + \tfrac{993563 y z}{1200} - \tfrac{117133 y}{400} - 242 z^{3} + \tfrac{2351 z^{2}}{4} - \tfrac{1813 z}{4} + \tfrac{215}{2}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{22}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,\tfrac{1}{2})\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{22} = \left(\begin{array}{c}\displaystyle \begin{cases} \tfrac{z \left(162000 x^{2} + 475472 x y + 318328 x z - 220725 x + 72000 y^{2} - 129736 y z - 97217 y - 401264 z^{2} + 131642 z + 15525\right)}{3600}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\45 x^{2} y + \tfrac{133909 x y^{2}}{450} - \tfrac{17342 x y z}{225} - \tfrac{917 x y}{16} - 4 x z - \tfrac{97217 y^{3}}{450} + \tfrac{15421 y^{2} z}{225} + \tfrac{113383 y^{2}}{3600} + 20 y z^{2} - \tfrac{32279 y z}{1800} + \tfrac{5 y}{16} - 4 z^{2} + 4 z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{101 x^{3}}{2} + \tfrac{34217 x^{2} y}{450} - \tfrac{6421 x^{2} z}{225} - \tfrac{365 x^{2}}{16} - \tfrac{10817 x y}{3600} + \tfrac{3721 x z}{1800} + \tfrac{5 x}{16} + 20 y^{2} z + 20 y z^{2} - 24 y z - 4 z^{2} + 4 z&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{641 x^{3}}{2} + \tfrac{107683 x^{2} y}{150} + \tfrac{77371 x^{2} z}{75} - \tfrac{12387 x^{2}}{16} + \tfrac{243607 x y^{2}}{450} + 1541 x y z - \tfrac{682519 x y}{600} + \tfrac{431843 x z^{2}}{450} - \tfrac{1882687 x z}{1200} + \tfrac{4791 x}{8} + \tfrac{64783 y^{3}}{450} + \tfrac{129704 y^{2} z}{225} - \tfrac{1549511 y^{2}}{3600} + \tfrac{306467 y z^{2}}{450} - \tfrac{17969 y z}{16} + \tfrac{86329 y}{200} + \tfrac{55921 z^{3}}{225} - \tfrac{1174757 z^{2}}{1800} + \tfrac{219717 z}{400} - \tfrac{2323}{16}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(- 547472 x y + 57736 x z + 61217 x - 306000 y^{2} - 296728 y z + 335025 y + 365264 z^{2} - 123542 z - 22725\right)}{3600}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 489736 x y + 61217 x - 232200 y^{2} - 5264 y z + 210825 y + 658 z - 22725\right)}{3600}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(345736 x^{2} - 761672 x y + 77264 x z - 62983 x - 306000 y^{2} - 82528 y z + 335025 y + 658 z - 22725\right)}{3600}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{28783 x^{3}}{450} - \tfrac{51907 x^{2} y}{450} - \tfrac{41954 x^{2} z}{225} + \tfrac{563111 x^{2}}{3600} - \tfrac{15883 x y^{2}}{150} - 349 x y z + \tfrac{141469 x y}{600} - \tfrac{81467 x z^{2}}{450} + \tfrac{5101 x z}{16} - \tfrac{26329 x}{200} - \tfrac{109 y^{3}}{2} - \tfrac{17221 y^{2} z}{75} + \tfrac{2343 y^{2}}{16} - \tfrac{105143 y z^{2}}{450} + \tfrac{445087 y z}{1200} - \tfrac{1049 y}{8} - \tfrac{13171 z^{3}}{225} + \tfrac{292307 z^{2}}{1800} - \tfrac{57217 z}{400} + \tfrac{627}{16}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(111736 x z - 13967 x + 68264 y z - 8533 y - 7200 z^{2} - 42300 z + 5400\right)}{3600}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 543736 x y + 655472 x z - 13967 x + 291736 y^{2} - 374672 y z + 92267 y + 144000 z^{2} - 143100 z + 5400\right)}{3600}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(- 255736 x^{2} + 3736 x y + 216272 x z + 86833 x + 64528 y z - 8533 y + 144000 z^{2} - 143100 z + 5400\right)}{3600}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{103967 x^{3}}{450} - \tfrac{243467 x^{2} y}{450} - \tfrac{190384 x^{2} z}{225} + \tfrac{2066239 x^{2}}{3600} - \tfrac{175033 x y^{2}}{450} - 1224 x y z + \tfrac{3369 x y}{4} - \tfrac{128567 x z^{2}}{150} + \tfrac{1569337 x z}{1200} - \tfrac{180967 x}{400} - \tfrac{35533 y^{3}}{450} - \tfrac{85016 y^{2} z}{225} + \tfrac{965861 y^{2}}{3600} - \tfrac{81133 y z^{2}}{150} + \tfrac{1005563 y z}{1200} - \tfrac{119533 y}{400} - 242 z^{3} + \tfrac{2367 z^{2}}{4} - \tfrac{1837 z}{4} + \tfrac{219}{2}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{23}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{2},0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{23} = \left(\begin{array}{c}\displaystyle \begin{cases} 45 x^{2} z + \tfrac{29717 x y z}{225} - 4 x y + \tfrac{39791 x z^{2}}{450} - \tfrac{917 x z}{16} + 20 y^{2} z - 4 y^{2} - \tfrac{16217 y z^{2}}{450} - \tfrac{111617 y z}{3600} + 4 y - \tfrac{25079 z^{3}}{225} + \tfrac{80221 z^{2}}{1800} + \tfrac{5 z}{16}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(162000 x^{2} + 1071272 x y - 277472 x z - 220725 x - 777736 y^{2} + 246736 y z + 84583 y + 72000 z^{2} - 50158 z + 15525\right)}{3600}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{101 x^{3}}{2} + \tfrac{34217 x^{2} y}{450} - \tfrac{6421 x^{2} z}{225} - \tfrac{365 x^{2}}{16} - \tfrac{39617 x y}{3600} + \tfrac{18121 x z}{1800} + \tfrac{5 x}{16} + 20 y^{2} z - 4 y^{2} + 20 y z^{2} - 24 y z + 4 y&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{641 x^{3}}{2} + \tfrac{107683 x^{2} y}{150} + \tfrac{77371 x^{2} z}{75} - \tfrac{12387 x^{2}}{16} + \tfrac{243607 x y^{2}}{450} + 1541 x y z - \tfrac{682519 x y}{600} + \tfrac{431843 x z^{2}}{450} - \tfrac{1882687 x z}{1200} + \tfrac{4791 x}{8} + \tfrac{64783 y^{3}}{450} + \tfrac{129704 y^{2} z}{225} - \tfrac{1549511 y^{2}}{3600} + \tfrac{306467 y z^{2}}{450} - \tfrac{17969 y z}{16} + \tfrac{86329 y}{200} + \tfrac{55921 z^{3}}{225} - \tfrac{1174757 z^{2}}{1800} + \tfrac{219717 z}{400} - \tfrac{2323}{16}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(- 547472 x y + 57736 x z + 61217 x - 306000 y^{2} - 296728 y z + 335025 y + 365264 z^{2} - 123542 z - 22725\right)}{3600}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 489736 x y + 61217 x - 232200 y^{2} - 5264 y z + 210825 y + 658 z - 22725\right)}{3600}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(345736 x^{2} - 761672 x y + 77264 x z - 62983 x - 306000 y^{2} - 82528 y z + 335025 y + 658 z - 22725\right)}{3600}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{28783 x^{3}}{450} - \tfrac{51907 x^{2} y}{450} - \tfrac{41954 x^{2} z}{225} + \tfrac{563111 x^{2}}{3600} - \tfrac{15883 x y^{2}}{150} - 349 x y z + \tfrac{141469 x y}{600} - \tfrac{81467 x z^{2}}{450} + \tfrac{5101 x z}{16} - \tfrac{26329 x}{200} - \tfrac{109 y^{3}}{2} - \tfrac{17221 y^{2} z}{75} + \tfrac{2343 y^{2}}{16} - \tfrac{105143 y z^{2}}{450} + \tfrac{445087 y z}{1200} - \tfrac{1049 y}{8} - \tfrac{13171 z^{3}}{225} + \tfrac{292307 z^{2}}{1800} - \tfrac{57217 z}{400} + \tfrac{627}{16}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(111736 x z - 13967 x + 68264 y z - 8533 y - 7200 z^{2} - 42300 z + 5400\right)}{3600}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 543736 x y + 655472 x z - 13967 x + 291736 y^{2} - 374672 y z + 92267 y + 144000 z^{2} - 143100 z + 5400\right)}{3600}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(- 255736 x^{2} + 3736 x y + 216272 x z + 86833 x + 64528 y z - 8533 y + 144000 z^{2} - 143100 z + 5400\right)}{3600}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{103967 x^{3}}{450} - \tfrac{243467 x^{2} y}{450} - \tfrac{190384 x^{2} z}{225} + \tfrac{2066239 x^{2}}{3600} - \tfrac{175033 x y^{2}}{450} - 1224 x y z + \tfrac{3369 x y}{4} - \tfrac{128567 x z^{2}}{150} + \tfrac{1569337 x z}{1200} - \tfrac{180967 x}{400} - \tfrac{35533 y^{3}}{450} - \tfrac{85016 y^{2} z}{225} + \tfrac{965861 y^{2}}{3600} - \tfrac{81133 y z^{2}}{150} + \tfrac{1005563 y z}{1200} - \tfrac{119533 y}{400} - 242 z^{3} + \tfrac{2367 z^{2}}{4} - \tfrac{1837 z}{4} + \tfrac{219}{2}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{24}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},0,\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\-1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{24} = \left(\begin{array}{c}\displaystyle \begin{cases} \tfrac{z \left(- 160 x^{2} - 288 x y - 24 x z + 159 x + 384 y z - 12 y - 104 z^{2} - 75 z - 6\right)}{4}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\- 40 x^{2} y + 50 x y^{2} - 128 x y z + \tfrac{151 x y}{4} + 2 x z + 16 y^{3} + 54 y^{2} z - 29 y^{2} + \tfrac{29 y z}{4} - \tfrac{3 y}{2}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(8 x^{2} + 96 x y - 296 x z + 47 x - 12 y + 37 z - 6\right)}{4}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\242 x^{3} + 692 x^{2} y + 706 x^{2} z - \tfrac{2351 x^{2}}{4} + 626 x y^{2} + 1224 x y z - \tfrac{4299 x y}{4} + 598 x z^{2} - 1055 x z + \tfrac{1813 x}{4} + 176 y^{3} + 486 y^{2} z - 455 y^{2} + 444 y z^{2} - \tfrac{3345 y z}{4} + \tfrac{773 y}{2} + 134 z^{3} - \tfrac{1525 z^{2}}{4} + \tfrac{1419 z}{4} - \tfrac{215}{2}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(- 180000 x^{2} + 194096 x y - 367048 x z + 156619 x + 86096 y^{2} - 950856 y z + 30023 y + 957904 z^{2} - 89214 z - 8119\right)}{9000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\- 20 x^{2} z + \tfrac{881 x y^{2}}{1125} + \tfrac{12619 x y}{9000} - 20 x z^{2} + 16 x z - \tfrac{26119 y^{3}}{1125} + \tfrac{37762 y^{2} z}{1125} - \tfrac{3881 y^{2}}{1000} - \tfrac{12131 y z}{4500} - \tfrac{8119 y}{9000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(1252952 x^{2} - 1540952 x y - 662096 x z - 52333 x + 86096 y^{2} + 784192 y z + 30023 y - 180000 z^{2} + 119738 z - 8119\right)}{9000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{248381 x^{3}}{1125} - \tfrac{197081 x^{2} y}{225} - \tfrac{827024 x^{2} z}{1125} + \tfrac{220661 x^{2}}{360} - \tfrac{1189667 x y^{2}}{1125} - \tfrac{2163334 x y z}{1125} + \tfrac{7242871 x y}{4500} - \tfrac{172781 x z^{2}}{225} + \tfrac{11984027 x z}{9000} - \tfrac{2538203 x}{4500} - \tfrac{150881 y^{3}}{375} - \tfrac{260062 y^{2} z}{225} + \tfrac{2893739 y^{2}}{3000} - \tfrac{125881 y z^{2}}{125} + \tfrac{3140563 y z}{1800} - \tfrac{1101167 y}{1500} - \tfrac{285262 z^{3}}{1125} + \tfrac{1023917 z^{2}}{1500} - \tfrac{1081667 z}{1800} + \tfrac{171881}{1000}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(262952 x z - 32869 x + 237904 y z - 29738 y + 334952 z^{2} - 250821 z + 26119\right)}{9000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\- \tfrac{55369 x y^{2}}{1125} + \tfrac{88238 x y z}{1125} - \tfrac{50869 x y}{9000} + 2 x z + \tfrac{15262 y^{3}}{1125} + \tfrac{7369 y^{2} z}{375} + \tfrac{8869 y^{2}}{1500} + \tfrac{34238 y z^{2}}{1125} - \tfrac{333773 y z}{9000} + \tfrac{26119 y}{9000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(- 1162952 x^{2} + 842096 x y + 1486952 x z + 32083 x - 604192 y z - 29738 y + 273904 z^{2} - 315773 z + 26119\right)}{9000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{34631 x^{3}}{1125} + \tfrac{264524 x^{2} y}{1125} + \tfrac{33731 x^{2} z}{225} - \tfrac{45611 x^{2}}{360} + \tfrac{85031 x y^{2}}{225} + \tfrac{822334 x y z}{1125} - \tfrac{4871777 x y}{9000} + \tfrac{287417 x z^{2}}{1125} - \tfrac{1860871 x z}{4500} + \tfrac{368539 x}{2250} + \tfrac{195262 y^{3}}{1125} + \tfrac{72631 y^{2} z}{125} - \tfrac{621917 y^{2}}{1500} + \tfrac{122362 y z^{2}}{225} - \tfrac{1528213 y z}{1800} + \tfrac{556067 y}{1800} + \tfrac{51131 z^{3}}{375} - \tfrac{1010489 z^{2}}{3000} + \tfrac{201271 z}{750} - \tfrac{67881}{1000}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{25}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\-1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{25} = \left(\begin{array}{c}\displaystyle \begin{cases} \tfrac{z \left(- 160 x^{2} - 288 x y - 24 x z + 159 x + 384 y z - 12 y - 104 z^{2} - 75 z - 6\right)}{4}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 160 x^{2} + 200 x y - 512 x z + 159 x + 64 y^{2} + 216 y z - 124 y + 37 z - 6\right)}{4}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(8 x^{2} + 96 x y - 296 x z + 47 x - 12 y + 37 z - 6\right)}{4}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\242 x^{3} + 692 x^{2} y + 706 x^{2} z - \tfrac{2367 x^{2}}{4} + 626 x y^{2} + 1224 x y z - \tfrac{4323 x y}{4} + 598 x z^{2} - 1065 x z + \tfrac{1837 x}{4} + 176 y^{3} + 486 y^{2} z - 457 y^{2} + 444 y z^{2} - \tfrac{3369 y z}{4} + \tfrac{781 y}{2} + 134 z^{3} - \tfrac{1541 z^{2}}{4} + \tfrac{1443 z}{4} - \tfrac{219}{2}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(- 180000 x^{2} + 194096 x y - 367048 x z + 156619 x + 86096 y^{2} - 950856 y z + 30023 y + 957904 z^{2} - 89214 z - 8119\right)}{9000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\- 20 x^{2} z + \tfrac{881 x y^{2}}{1125} - \tfrac{59381 x y}{9000} - 20 x z^{2} + 24 x z - \tfrac{26119 y^{3}}{1125} + \tfrac{37762 y^{2} z}{1125} - \tfrac{7881 y^{2}}{1000} - \tfrac{12131 y z}{4500} + \tfrac{27881 y}{9000} + 4 z^{2} - 4 z&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{156619 x^{3}}{1125} - \tfrac{192619 x^{2} y}{1125} - \tfrac{82762 x^{2} z}{1125} - \tfrac{124333 x^{2}}{9000} + \tfrac{10762 x y^{2}}{1125} + \tfrac{98024 x y z}{1125} - \tfrac{5977 x y}{9000} - 20 x z^{2} + \tfrac{77869 x z}{4500} + \tfrac{27881 x}{9000} + 4 y z + 4 z^{2} - 4 z&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{248381 x^{3}}{1125} - \tfrac{197081 x^{2} y}{225} - \tfrac{827024 x^{2} z}{1125} + \tfrac{223541 x^{2}}{360} - \tfrac{1189667 x y^{2}}{1125} - \tfrac{2163334 x y z}{1125} + \tfrac{7296871 x y}{4500} - \tfrac{172781 x z^{2}}{225} + \tfrac{12164027 x z}{9000} - \tfrac{2592203 x}{4500} - \tfrac{150881 y^{3}}{375} - \tfrac{260062 y^{2} z}{225} + \tfrac{2905739 y^{2}}{3000} - \tfrac{125881 y z^{2}}{125} + \tfrac{3162163 y z}{1800} - \tfrac{1113167 y}{1500} - \tfrac{285262 z^{3}}{1125} + \tfrac{1035917 z^{2}}{1500} - \tfrac{1103267 z}{1800} + \tfrac{175881}{1000}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(262952 x z - 32869 x + 237904 y z - 29738 y + 334952 z^{2} - 250821 z + 26119\right)}{9000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 442952 x y + 705904 x z - 32869 x + 122096 y^{2} + 176856 y z + 35214 y + 273904 z^{2} - 315773 z + 26119\right)}{9000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(- 1162952 x^{2} + 842096 x y + 1486952 x z + 32083 x - 604192 y z - 29738 y + 273904 z^{2} - 315773 z + 26119\right)}{9000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{34631 x^{3}}{1125} + \tfrac{264524 x^{2} y}{1125} + \tfrac{33731 x^{2} z}{225} - \tfrac{47051 x^{2}}{360} + \tfrac{85031 x y^{2}}{225} + \tfrac{822334 x y z}{1125} - \tfrac{4925777 x y}{9000} + \tfrac{287417 x z^{2}}{1125} - \tfrac{1905871 x z}{4500} + \tfrac{382039 x}{2250} + \tfrac{195262 y^{3}}{1125} + \tfrac{72631 y^{2} z}{125} - \tfrac{624917 y^{2}}{1500} + \tfrac{122362 y z^{2}}{225} - \tfrac{1539013 y z}{1800} + \tfrac{563267 y}{1800} + \tfrac{51131 z^{3}}{375} - \tfrac{1022489 z^{2}}{3000} + \tfrac{205771 z}{750} - \tfrac{69881}{1000}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{26}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},0,0)\cdot\left(\begin{array}{c}0\\-1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{26} = \left(\begin{array}{c}\displaystyle \begin{cases} \tfrac{z \left(- 160 x^{2} - 288 x y - 24 x z + 159 x + 384 y z - 12 y - 104 z^{2} - 75 z - 6\right)}{4}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 160 x^{2} + 200 x y - 512 x z + 159 x + 64 y^{2} + 216 y z - 124 y + 37 z - 6\right)}{4}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(8 x^{2} + 96 x y - 296 x z + 47 x - 12 y + 37 z - 6\right)}{4}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\242 x^{3} + 692 x^{2} y + 706 x^{2} z - \tfrac{2367 x^{2}}{4} + 626 x y^{2} + 1224 x y z - \tfrac{4323 x y}{4} + 598 x z^{2} - 1065 x z + \tfrac{1837 x}{4} + 176 y^{3} + 486 y^{2} z - 457 y^{2} + 444 y z^{2} - \tfrac{3369 y z}{4} + \tfrac{781 y}{2} + 134 z^{3} - \tfrac{1541 z^{2}}{4} + \tfrac{1443 z}{4} - \tfrac{219}{2}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} - 20 x^{2} z + 4 x^{2} + \tfrac{24262 x y z}{1125} + 4 x y - \tfrac{45881 x z^{2}}{1125} + \tfrac{192619 x z}{9000} - 4 x + \tfrac{10762 y^{2} z}{1125} - \tfrac{39619 y z^{2}}{375} - \tfrac{5977 y z}{9000} + \tfrac{119738 z^{3}}{1125} - \tfrac{26869 z^{2}}{1500} + \tfrac{27881 z}{9000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\- 20 x^{2} z + 4 x^{2} + \tfrac{881 x y^{2}}{1125} + \tfrac{12619 x y}{9000} - 20 x z^{2} + 24 x z - 4 x - \tfrac{26119 y^{3}}{1125} + \tfrac{37762 y^{2} z}{1125} - \tfrac{7881 y^{2}}{1000} - \tfrac{48131 y z}{4500} + \tfrac{27881 y}{9000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(1252952 x^{2} - 1540952 x y - 662096 x z - 52333 x + 86096 y^{2} + 784192 y z + 30023 y - 180000 z^{2} + 119738 z - 8119\right)}{9000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{248381 x^{3}}{1125} - \tfrac{197081 x^{2} y}{225} - \tfrac{827024 x^{2} z}{1125} + \tfrac{223541 x^{2}}{360} - \tfrac{1189667 x y^{2}}{1125} - \tfrac{2163334 x y z}{1125} + \tfrac{7296871 x y}{4500} - \tfrac{172781 x z^{2}}{225} + \tfrac{12164027 x z}{9000} - \tfrac{2592203 x}{4500} - \tfrac{150881 y^{3}}{375} - \tfrac{260062 y^{2} z}{225} + \tfrac{2905739 y^{2}}{3000} - \tfrac{125881 y z^{2}}{125} + \tfrac{3162163 y z}{1800} - \tfrac{1113167 y}{1500} - \tfrac{285262 z^{3}}{1125} + \tfrac{1035917 z^{2}}{1500} - \tfrac{1103267 z}{1800} + \tfrac{175881}{1000}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(262952 x z - 32869 x + 237904 y z - 29738 y + 334952 z^{2} - 250821 z + 26119\right)}{9000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 442952 x y + 705904 x z - 32869 x + 122096 y^{2} + 176856 y z + 35214 y + 273904 z^{2} - 315773 z + 26119\right)}{9000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(- 1162952 x^{2} + 842096 x y + 1486952 x z + 32083 x - 604192 y z - 29738 y + 273904 z^{2} - 315773 z + 26119\right)}{9000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{34631 x^{3}}{1125} + \tfrac{264524 x^{2} y}{1125} + \tfrac{33731 x^{2} z}{225} - \tfrac{47051 x^{2}}{360} + \tfrac{85031 x y^{2}}{225} + \tfrac{822334 x y z}{1125} - \tfrac{4925777 x y}{9000} + \tfrac{287417 x z^{2}}{1125} - \tfrac{1905871 x z}{4500} + \tfrac{382039 x}{2250} + \tfrac{195262 y^{3}}{1125} + \tfrac{72631 y^{2} z}{125} - \tfrac{624917 y^{2}}{1500} + \tfrac{122362 y z^{2}}{225} - \tfrac{1539013 y z}{1800} + \tfrac{563267 y}{1800} + \tfrac{51131 z^{3}}{375} - \tfrac{1022489 z^{2}}{3000} + \tfrac{205771 z}{750} - \tfrac{69881}{1000}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{27}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},\tfrac{1}{2},0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{27} = \left(\begin{array}{c}\displaystyle \begin{cases} - \tfrac{171562 x^{2} z}{1125} - \tfrac{210937 x y z}{1125} - 2 x y - \tfrac{80906 x z^{2}}{375} + \tfrac{1501027 x z}{9000} + \tfrac{233437 y z^{2}}{2250} + \tfrac{224437 y z}{18000} + \tfrac{258749 z^{3}}{2250} - \tfrac{305249 z^{2}}{6000} - \tfrac{94781 z}{9000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 2744992 x^{2} - 3838496 x y - 3419984 x z + 2966054 x + 3307496 y^{2} + 629992 y z - 1040059 y + 348749 z - 189562\right)}{18000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(- 1768496 x^{2} - 1507496 x y - 2789992 x z + 1737558 x + 188437 y + 348749 z - 189562\right)}{18000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{17531 x^{3}}{375} + \tfrac{342559 x^{2} y}{2250} + \tfrac{38699 x^{2} z}{450} - \tfrac{294089 x^{2}}{3000} + \tfrac{29081 x y^{2}}{225} + \tfrac{138934 x y z}{1125} - \tfrac{3334459 x y}{18000} - \tfrac{719 x z^{2}}{125} - \tfrac{198551 x z}{3600} + \tfrac{19523 x}{375} + \tfrac{53437 y^{3}}{2250} + \tfrac{5623 y^{2} z}{2250} - \tfrac{36157 y^{2}}{720} - \tfrac{29813 y z^{2}}{450} + \tfrac{271423 y z}{9000} + \tfrac{490487 y}{18000} - \tfrac{101251 z^{3}}{2250} + \tfrac{482257 z^{2}}{6000} - \tfrac{124541 z}{3600} - \tfrac{781}{1000}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 115 x y z - 2 x y - \tfrac{95 x z^{2}}{2} - \tfrac{103 x z}{16} + 40 y^{2} z - 37 y z^{2} - \tfrac{151 y z}{4} - \tfrac{45 z^{3}}{2} + \tfrac{451 z^{2}}{16} + \tfrac{3 z}{2}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(1080 x y - 135 x - 32 y^{2} - 280 y z - 188 y + 35 z + 24\right)}{16}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(520 x^{2} - 112 x y - 1640 x z + 313 x + 640 y^{2} + 1360 y z - 636 y + 35 z + 24\right)}{16}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{255 x^{3}}{2} - 572 x^{2} y - \tfrac{875 x^{2} z}{2} + \tfrac{5879 x^{2}}{16} - \tfrac{1373 x y^{2}}{2} - 1224 x y z + \tfrac{16451 x y}{16} - \tfrac{985 x z^{2}}{2} + \tfrac{3345 x z}{4} - \tfrac{5559 x}{16} - 242 y^{3} - \tfrac{1423 y^{2} z}{2} + \tfrac{2351 y^{2}}{4} - 652 y z^{2} + \tfrac{17625 y z}{16} - \tfrac{1813 y}{4} - \tfrac{365 z^{3}}{2} + \tfrac{7501 z^{2}}{16} - \tfrac{6301 z}{16} + \tfrac{215}{2}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 20 x^{2} y + 20 x y^{2} - 16 x y + \tfrac{213749 x z^{2}}{2250} - \tfrac{240749 x z}{18000} + \tfrac{120937 y z^{2}}{2250} - \tfrac{147937 y z}{18000} + \tfrac{94781 z^{3}}{1125} - \tfrac{64781 z^{2}}{1000} + \tfrac{76781 z}{9000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(360000 x^{2} - 989992 x y + 3059984 x z - 528749 x - 3487496 y^{2} + 3946496 y z + 1080559 y + 2024992 z^{2} - 2394554 z + 153562\right)}{18000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(- 4229992 x^{2} - 247496 x y + 5431488 x z + 987747 x + 360000 y^{2} + 1574992 y z - 435937 y + 2024992 z^{2} - 2394554 z + 153562\right)}{18000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{281251 x^{3}}{2250} + \tfrac{205313 x^{2} y}{450} + \tfrac{53969 x^{2} z}{125} - \tfrac{2090257 x^{2}}{6000} + \tfrac{1119377 x y^{2}}{2250} + \tfrac{1202066 x y z}{1125} - \tfrac{7383673 x y}{9000} + \tfrac{236701 x z^{2}}{450} - \tfrac{3026149 x z}{3600} + \tfrac{1175741 x}{3600} + \tfrac{374063 y^{3}}{2250} + \tfrac{134269 y^{2} z}{225} - \tfrac{313943 y^{2}}{720} + \tfrac{1461941 y z^{2}}{2250} - \tfrac{18193541 y z}{18000} + \tfrac{6714013 y}{18000} + \tfrac{82219 z^{3}}{375} - \tfrac{1589161 z^{2}}{3000} + \tfrac{620533 z}{1500} - \tfrac{103219}{1000}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{28}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{2},0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{28} = \left(\begin{array}{c}\displaystyle \begin{cases} \tfrac{z \left(- 2744992 x^{2} - 3374992 x y - 3883488 x z + 2966054 x + 1867496 y z + 188437 y + 2069992 z^{2} - 879747 z - 189562\right)}{18000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 2744992 x^{2} - 3838496 x y - 3419984 x z + 2966054 x + 3307496 y^{2} + 629992 y z - 1040059 y + 348749 z - 189562\right)}{18000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(- 1768496 x^{2} - 1507496 x y - 2789992 x z + 1737558 x + 188437 y + 348749 z - 189562\right)}{18000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{17531 x^{3}}{375} + \tfrac{342559 x^{2} y}{2250} + \tfrac{38699 x^{2} z}{450} - \tfrac{282089 x^{2}}{3000} + \tfrac{29081 x y^{2}}{225} + \tfrac{138934 x y z}{1125} - \tfrac{3154459 x y}{18000} - \tfrac{719 x z^{2}}{125} - \tfrac{176951 x z}{3600} + \tfrac{17273 x}{375} + \tfrac{53437 y^{3}}{2250} + \tfrac{5623 y^{2} z}{2250} - \tfrac{33277 y^{2}}{720} - \tfrac{29813 y z^{2}}{450} + \tfrac{325423 y z}{9000} + \tfrac{382487 y}{18000} - \tfrac{101251 z^{3}}{2250} + \tfrac{494257 z^{2}}{6000} - \tfrac{138941 z}{3600} + \tfrac{1219}{1000}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(1840 x y - 760 x z - 135 x + 640 y^{2} - 592 y z - 636 y - 360 z^{2} + 483 z + 24\right)}{16}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(1080 x y - 135 x - 32 y^{2} - 280 y z - 188 y + 35 z + 24\right)}{16}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(520 x^{2} - 112 x y - 1640 x z + 313 x + 640 y^{2} + 1360 y z - 636 y + 35 z + 24\right)}{16}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{255 x^{3}}{2} - 572 x^{2} y - \tfrac{875 x^{2} z}{2} + \tfrac{5943 x^{2}}{16} - \tfrac{1373 x y^{2}}{2} - 1224 x y z + \tfrac{16611 x y}{16} - \tfrac{985 x z^{2}}{2} + \tfrac{3369 x z}{4} - \tfrac{5655 x}{16} - 242 y^{3} - \tfrac{1423 y^{2} z}{2} + \tfrac{2367 y^{2}}{4} - 652 y z^{2} + \tfrac{17721 y z}{16} - \tfrac{1837 y}{4} - \tfrac{365 z^{3}}{2} + \tfrac{7533 z^{2}}{16} - \tfrac{6365 z}{16} + \tfrac{219}{2}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 20 x^{2} y + 20 x y^{2} - 24 x y + \tfrac{213749 x z^{2}}{2250} - \tfrac{96749 x z}{18000} - 4 y^{2} + \tfrac{120937 y z^{2}}{2250} - \tfrac{147937 y z}{18000} + 4 y + \tfrac{94781 z^{3}}{1125} - \tfrac{60781 z^{2}}{1000} + \tfrac{40781 z}{9000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(360000 x^{2} - 989992 x y + 3059984 x z - 528749 x - 3487496 y^{2} + 3946496 y z + 1080559 y + 2024992 z^{2} - 2394554 z + 153562\right)}{18000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{528749 x^{3}}{2250} - \tfrac{30937 x^{2} y}{2250} + \tfrac{113156 x^{2} z}{375} + \tfrac{377249 x^{2}}{6000} + 20 x y^{2} + \tfrac{98437 x y z}{1125} - \tfrac{507937 x y}{18000} + \tfrac{126562 x z^{2}}{1125} - \tfrac{1161277 x z}{9000} + \tfrac{40781 x}{9000} - 4 y^{2} - 4 y z + 4 y&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{281251 x^{3}}{2250} + \tfrac{205313 x^{2} y}{450} + \tfrac{53969 x^{2} z}{125} - \tfrac{2138257 x^{2}}{6000} + \tfrac{1119377 x y^{2}}{2250} + \tfrac{1202066 x y z}{1125} - \tfrac{7563673 x y}{9000} + \tfrac{236701 x z^{2}}{450} - \tfrac{3069349 x z}{3600} + \tfrac{1218941 x}{3600} + \tfrac{374063 y^{3}}{2250} + \tfrac{134269 y^{2} z}{225} - \tfrac{319703 y^{2}}{720} + \tfrac{1461941 y z^{2}}{2250} - \tfrac{18409541 y z}{18000} + \tfrac{6930013 y}{18000} + \tfrac{82219 z^{3}}{375} - \tfrac{1601161 z^{2}}{3000} + \tfrac{632533 z}{1500} - \tfrac{107219}{1000}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{29}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},0,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{29} = \left(\begin{array}{c}\displaystyle \begin{cases} \tfrac{z \left(- 2744992 x^{2} - 3374992 x y - 3883488 x z + 2966054 x + 1867496 y z + 188437 y + 2069992 z^{2} - 879747 z - 189562\right)}{18000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 2744992 x^{2} - 3838496 x y - 3419984 x z + 2966054 x + 3307496 y^{2} + 629992 y z - 1040059 y + 348749 z - 189562\right)}{18000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(- 1768496 x^{2} - 1507496 x y - 2789992 x z + 1737558 x + 188437 y + 348749 z - 189562\right)}{18000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{17531 x^{3}}{375} + \tfrac{342559 x^{2} y}{2250} + \tfrac{38699 x^{2} z}{450} - \tfrac{282089 x^{2}}{3000} + \tfrac{29081 x y^{2}}{225} + \tfrac{138934 x y z}{1125} - \tfrac{3154459 x y}{18000} - \tfrac{719 x z^{2}}{125} - \tfrac{176951 x z}{3600} + \tfrac{17273 x}{375} + \tfrac{53437 y^{3}}{2250} + \tfrac{5623 y^{2} z}{2250} - \tfrac{33277 y^{2}}{720} - \tfrac{29813 y z^{2}}{450} + \tfrac{325423 y z}{9000} + \tfrac{382487 y}{18000} - \tfrac{101251 z^{3}}{2250} + \tfrac{494257 z^{2}}{6000} - \tfrac{138941 z}{3600} + \tfrac{1219}{1000}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(1840 x y - 760 x z - 135 x + 640 y^{2} - 592 y z - 636 y - 360 z^{2} + 483 z + 24\right)}{16}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(1080 x y - 135 x - 32 y^{2} - 280 y z - 188 y + 35 z + 24\right)}{16}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(520 x^{2} - 112 x y - 1640 x z + 313 x + 640 y^{2} + 1360 y z - 636 y + 35 z + 24\right)}{16}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{255 x^{3}}{2} - 572 x^{2} y - \tfrac{875 x^{2} z}{2} + \tfrac{5943 x^{2}}{16} - \tfrac{1373 x y^{2}}{2} - 1224 x y z + \tfrac{16611 x y}{16} - \tfrac{985 x z^{2}}{2} + \tfrac{3369 x z}{4} - \tfrac{5655 x}{16} - 242 y^{3} - \tfrac{1423 y^{2} z}{2} + \tfrac{2367 y^{2}}{4} - 652 y z^{2} + \tfrac{17721 y z}{16} - \tfrac{1837 y}{4} - \tfrac{365 z^{3}}{2} + \tfrac{7533 z^{2}}{16} - \tfrac{6365 z}{16} + \tfrac{219}{2}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 20 x^{2} y - 4 x^{2} + 20 x y^{2} - 24 x y + \tfrac{213749 x z^{2}}{2250} - \tfrac{240749 x z}{18000} + 4 x + \tfrac{120937 y z^{2}}{2250} - \tfrac{3937 y z}{18000} + \tfrac{94781 z^{3}}{1125} - \tfrac{60781 z^{2}}{1000} + \tfrac{40781 z}{9000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\20 x^{2} y - 4 x^{2} - \tfrac{123749 x y^{2}}{2250} + \tfrac{191249 x y z}{1125} - \tfrac{600749 x y}{18000} - 4 x z + 4 x - \tfrac{435937 y^{3}}{2250} + \tfrac{246656 y^{2} z}{1125} + \tfrac{1224559 y^{2}}{18000} + \tfrac{126562 y z^{2}}{1125} - \tfrac{1161277 y z}{9000} + \tfrac{40781 y}{9000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(- 4229992 x^{2} - 247496 x y + 5431488 x z + 987747 x + 360000 y^{2} + 1574992 y z - 435937 y + 2024992 z^{2} - 2394554 z + 153562\right)}{18000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{281251 x^{3}}{2250} + \tfrac{205313 x^{2} y}{450} + \tfrac{53969 x^{2} z}{125} - \tfrac{2138257 x^{2}}{6000} + \tfrac{1119377 x y^{2}}{2250} + \tfrac{1202066 x y z}{1125} - \tfrac{7563673 x y}{9000} + \tfrac{236701 x z^{2}}{450} - \tfrac{3069349 x z}{3600} + \tfrac{1218941 x}{3600} + \tfrac{374063 y^{3}}{2250} + \tfrac{134269 y^{2} z}{225} - \tfrac{319703 y^{2}}{720} + \tfrac{1461941 y z^{2}}{2250} - \tfrac{18409541 y z}{18000} + \tfrac{6930013 y}{18000} + \tfrac{82219 z^{3}}{375} - \tfrac{1601161 z^{2}}{3000} + \tfrac{632533 z}{1500} - \tfrac{107219}{1000}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{30}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{8},\tfrac{1}{8},\tfrac{1}{8})\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{30} = \left(\begin{array}{c}\displaystyle \begin{cases} 16 z \left(- x - y - 2 z + 1\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\16 y \left(- x - 2 y - z + 1\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\16 x \left(- 2 x - y - z + 1\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{31}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{8},\tfrac{1}{8},\tfrac{1}{8})\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{31} = \left(\begin{array}{c}\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 16 z \left(- x - y - 2 z + 1\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\16 y \left(- x - 2 y - z + 1\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\16 x \left(- 2 x - y - z + 1\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{32}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{8},\tfrac{1}{8},\tfrac{1}{8})\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{32} = \left(\begin{array}{c}\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 16 z \left(- x - y - 2 z + 1\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\16 y \left(- x - 2 y - z + 1\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\16 x \left(- 2 x - y - z + 1\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{33}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{5}{8},\tfrac{1}{8},\tfrac{1}{8})\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{33} = \left(\begin{array}{c}\displaystyle \begin{cases} 16 z \left(x - z\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\16 y \left(x - y\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- 32 x^{2} - 48 x y - 48 x z + 48 x - 16 y^{2} - 32 y z + 32 y - 16 z^{2} + 32 z - 16&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{34}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{5}{8},\tfrac{1}{8},\tfrac{1}{8})\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{34} = \left(\begin{array}{c}\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 16 z \left(x - z\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\16 y \left(x - y\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- 32 x^{2} - 48 x y - 48 x z + 48 x - 16 y^{2} - 32 y z + 32 y - 16 z^{2} + 32 z - 16&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{35}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{5}{8},\tfrac{1}{8},\tfrac{1}{8})\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{35} = \left(\begin{array}{c}\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 16 z \left(x - z\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\16 y \left(x - y\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- 32 x^{2} - 48 x y - 48 x z + 48 x - 16 y^{2} - 32 y z + 32 y - 16 z^{2} + 32 z - 16&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{36}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{8},\tfrac{5}{8},\tfrac{1}{8})\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{36} = \left(\begin{array}{c}\displaystyle \begin{cases} 16 z \left(y - z\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\16 x \left(- x + y\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- 16 x^{2} - 48 x y - 32 x z + 32 x - 32 y^{2} - 48 y z + 48 y - 16 z^{2} + 32 z - 16&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{37}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{8},\tfrac{5}{8},\tfrac{1}{8})\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{37} = \left(\begin{array}{c}\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 16 z \left(y - z\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\16 x \left(- x + y\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- 16 x^{2} - 48 x y - 32 x z + 32 x - 32 y^{2} - 48 y z + 48 y - 16 z^{2} + 32 z - 16&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{38}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{8},\tfrac{5}{8},\tfrac{1}{8})\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{38} = \left(\begin{array}{c}\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 16 z \left(y - z\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\16 x \left(- x + y\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- 16 x^{2} - 48 x y - 32 x z + 32 x - 32 y^{2} - 48 y z + 48 y - 16 z^{2} + 32 z - 16&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{39}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{8},\tfrac{1}{8},\tfrac{5}{8})\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{39} = \left(\begin{array}{c}\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\16 y \left(- y + z\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\16 x \left(- x + z\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- 16 x^{2} - 32 x y - 48 x z + 32 x - 16 y^{2} - 48 y z + 32 y - 32 z^{2} + 48 z - 16&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{40}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{8},\tfrac{1}{8},\tfrac{5}{8})\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{40} = \left(\begin{array}{c}\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\16 y \left(- y + z\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\16 x \left(- x + z\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- 16 x^{2} - 32 x y - 48 x z + 32 x - 16 y^{2} - 48 y z + 32 y - 32 z^{2} + 48 z - 16&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{41}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{8},\tfrac{1}{8},\tfrac{5}{8})\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{41} = \left(\begin{array}{c}\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\16 y \left(- y + z\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\16 x \left(- x + z\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- 16 x^{2} - 32 x y - 48 x z + 32 x - 16 y^{2} - 48 y z + 32 y - 32 z^{2} + 48 z - 16&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{42}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{0}\)
where \(f_{0}\) is the 0th face;
and \(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0.

\(\displaystyle \boldsymbol{\phi}_{42} = \left(\begin{array}{c}\displaystyle \begin{cases} \tfrac{z \left(- 1470000 x^{2} - 1151488 x y - 1779512 x z + 1468875 x + 1775744 y z - 78032 y + 3429256 z^{2} - 1692343 z + 25125\right)}{3000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 1470000 x^{2} - 872488 x y - 2058512 x z + 1468875 x + 2975744 y^{2} + 2229256 y z - 1749032 y - 21343 z + 25125\right)}{3000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(9000 x^{2} + 624256 x y + 170744 x z - 202125 x - 78032 y - 21343 z + 25125\right)}{3000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\263 x^{3} + \tfrac{91468 x^{2} y}{125} + \tfrac{148157 x^{2} z}{125} - \tfrac{5781 x^{2}}{8} + \tfrac{277747 x y^{2}}{375} + 2166 x y z - \tfrac{1371273 x y}{1000} + \tfrac{504503 x z^{2}}{375} - \tfrac{498713 x z}{250} + \tfrac{2593 x}{4} + \tfrac{101968 y^{3}}{375} + \tfrac{377593 y^{2} z}{375} - \tfrac{268057 y^{2}}{375} + \tfrac{434282 y z^{2}}{375} - \tfrac{14327 y z}{8} + \tfrac{631529 y}{1000} + \tfrac{158657 z^{3}}{375} - \tfrac{3108169 z^{2}}{3000} + \tfrac{200399 z}{250} - \tfrac{1509}{8}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(- 1308512 x y + 1854256 x z - 68218 x - 1470000 y^{2} - 1622488 y z + 1468875 y + 3350744 z^{2} - 1702157 z + 25125\right)}{3000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(545744 x y - 68218 x + 9000 y^{2} + 249256 y z - 202125 y - 31157 z + 25125\right)}{3000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(3054256 x^{2} - 1029512 x y + 2150744 x z - 1739218 x - 1470000 y^{2} - 1901488 y z + 1468875 y - 31157 z + 25125\right)}{3000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{111782 x^{3}}{375} + \tfrac{317003 x^{2} y}{375} + \tfrac{387407 x^{2} z}{375} - \tfrac{1155647 x^{2}}{1500} + \tfrac{101282 x y^{2}}{125} + 2166 x y z - \tfrac{1479227 x y}{1000} + \tfrac{424468 x z^{2}}{375} - \tfrac{14327 x z}{8} + \tfrac{660971 x}{1000} + 263 y^{3} + \tfrac{138343 y^{2} z}{125} - \tfrac{5781 y^{2}}{8} + \tfrac{465247 y z^{2}}{375} - \tfrac{943449 y z}{500} + \tfrac{2593 y}{4} + \tfrac{148843 z^{3}}{375} - \tfrac{2941331 z^{2}}{3000} + \tfrac{386077 z}{500} - \tfrac{1509}{8}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(1062128 x z - 132766 x + 1022872 y z - 127859 y + 567000 z^{2} - 733875 z + 82875\right)}{1500}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(137872 x y + 924256 x z - 132766 x + 777128 y^{2} + 422744 y z - 400859 y + 390000 z^{2} - 460875 z + 82875\right)}{1500}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(737872 x^{2} + 177128 x y + 501256 x z - 405766 x + 845744 y z - 127859 y + 390000 z^{2} - 460875 z + 82875\right)}{1500}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{85532 x^{3}}{375} - \tfrac{231782 x^{2} y}{375} - \tfrac{612878 x^{2} z}{375} + \tfrac{542011 x^{2}}{750} - \tfrac{221968 x y^{2}}{375} - 3084 x y z + \tfrac{5399 x y}{4} - \tfrac{283532 x z^{2}}{125} + \tfrac{1494051 x z}{500} - \tfrac{363923 x}{500} - \tfrac{75718 y^{3}}{375} - \tfrac{573622 y^{2} z}{375} + \tfrac{1000603 y^{2}}{1500} - \tfrac{273718 y z^{2}}{125} + \tfrac{720037 y z}{250} - \tfrac{174601 y}{250} - 862 z^{3} + \tfrac{7797 z^{2}}{4} - \tfrac{2641 z}{2} + \tfrac{933}{4}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{43}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{1}\)
where \(f_{1}\) is the 1st face;
and \(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1.

\(\displaystyle \boldsymbol{\phi}_{43} = \left(\begin{array}{c}\displaystyle \begin{cases} \tfrac{z \left(- 162000 x^{2} - 475472 x y - 318328 x z + 220725 x - 72000 y^{2} + 129736 y z + 97217 y + 401264 z^{2} - 131642 z - 15525\right)}{600}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 162000 x^{2} - 1071272 x y + 277472 x z + 220725 x + 777736 y^{2} - 246736 y z - 84583 y - 72000 z^{2} + 50158 z - 15525\right)}{600}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- 303 x^{3} - \tfrac{34217 x^{2} y}{75} + \tfrac{12842 x^{2} z}{75} + \tfrac{1479 x^{2}}{8} + \tfrac{25217 x y}{600} - \tfrac{10921 x z}{300} - \tfrac{207 x}{8} - 120 y^{2} z - 120 y z^{2} + 120 y z&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- 1923 x^{3} - \tfrac{107683 x^{2} y}{25} - \tfrac{154742 x^{2} z}{25} + \tfrac{37161 x^{2}}{8} - \tfrac{243607 x y^{2}}{75} - 9246 x y z + \tfrac{682519 x y}{100} - \tfrac{431843 x z^{2}}{75} + \tfrac{1882687 x z}{200} - \tfrac{14373 x}{4} - \tfrac{64783 y^{3}}{75} - \tfrac{259408 y^{2} z}{75} + \tfrac{1549511 y^{2}}{600} - \tfrac{306467 y z^{2}}{75} + \tfrac{53907 y z}{8} - \tfrac{258987 y}{100} - \tfrac{111842 z^{3}}{75} + \tfrac{1174757 z^{2}}{300} - \tfrac{659151 z}{200} + \tfrac{6969}{8}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(547472 x y - 57736 x z - 61217 x + 306000 y^{2} + 296728 y z - 335025 y - 365264 z^{2} + 123542 z + 22725\right)}{600}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(489736 x y - 61217 x + 232200 y^{2} + 5264 y z - 210825 y - 658 z + 22725\right)}{600}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(- 345736 x^{2} + 761672 x y - 77264 x z + 62983 x + 306000 y^{2} + 82528 y z - 335025 y - 658 z + 22725\right)}{600}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{28783 x^{3}}{75} + \tfrac{51907 x^{2} y}{75} + \tfrac{83908 x^{2} z}{75} - \tfrac{563111 x^{2}}{600} + \tfrac{15883 x y^{2}}{25} + 2094 x y z - \tfrac{141469 x y}{100} + \tfrac{81467 x z^{2}}{75} - \tfrac{15303 x z}{8} + \tfrac{78987 x}{100} + 327 y^{3} + \tfrac{34442 y^{2} z}{25} - \tfrac{7029 y^{2}}{8} + \tfrac{105143 y z^{2}}{75} - \tfrac{445087 y z}{200} + \tfrac{3147 y}{4} + \tfrac{26342 z^{3}}{75} - \tfrac{292307 z^{2}}{300} + \tfrac{171651 z}{200} - \tfrac{1881}{8}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(- 111736 x z + 13967 x - 68264 y z + 8533 y + 7200 z^{2} + 42300 z - 5400\right)}{600}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(543736 x y - 655472 x z + 13967 x - 291736 y^{2} + 374672 y z - 92267 y - 144000 z^{2} + 143100 z - 5400\right)}{600}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(255736 x^{2} - 3736 x y - 216272 x z - 86833 x - 64528 y z + 8533 y - 144000 z^{2} + 143100 z - 5400\right)}{600}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{103967 x^{3}}{75} + \tfrac{243467 x^{2} y}{75} + \tfrac{380768 x^{2} z}{75} - \tfrac{2066239 x^{2}}{600} + \tfrac{175033 x y^{2}}{75} + 7344 x y z - \tfrac{10107 x y}{2} + \tfrac{128567 x z^{2}}{25} - \tfrac{1569337 x z}{200} + \tfrac{542901 x}{200} + \tfrac{35533 y^{3}}{75} + \tfrac{170032 y^{2} z}{75} - \tfrac{965861 y^{2}}{600} + \tfrac{81133 y z^{2}}{25} - \tfrac{1005563 y z}{200} + \tfrac{358599 y}{200} + 1452 z^{3} - \tfrac{7101 z^{2}}{2} + \tfrac{5511 z}{2} - 657&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{44}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{2}\)
where \(f_{2}\) is the 2nd face;
and \(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2.

\(\displaystyle \boldsymbol{\phi}_{44} = \left(\begin{array}{c}\displaystyle \begin{cases} \tfrac{3 z \left(160 x^{2} + 288 x y + 24 x z - 159 x - 384 y z + 12 y + 104 z^{2} + 75 z + 6\right)}{2}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{3 y \left(160 x^{2} - 200 x y + 512 x z - 159 x - 64 y^{2} - 216 y z + 124 y - 37 z + 6\right)}{2}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{3 x \left(- 8 x^{2} - 96 x y + 296 x z - 47 x + 12 y - 37 z + 6\right)}{2}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- 1452 x^{3} - 4152 x^{2} y - 4236 x^{2} z + \tfrac{7101 x^{2}}{2} - 3756 x y^{2} - 7344 x y z + \tfrac{12969 x y}{2} - 3588 x z^{2} + 6390 x z - \tfrac{5511 x}{2} - 1056 y^{3} - 2916 y^{2} z + 2742 y^{2} - 2664 y z^{2} + \tfrac{10107 y z}{2} - 2343 y - 804 z^{3} + \tfrac{4623 z^{2}}{2} - \tfrac{4329 z}{2} + 657&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(180000 x^{2} - 194096 x y + 367048 x z - 156619 x - 86096 y^{2} + 950856 y z - 30023 y - 957904 z^{2} + 89214 z + 8119\right)}{1500}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\120 x^{2} z - \tfrac{1762 x y^{2}}{375} + \tfrac{23381 x y}{1500} + 120 x z^{2} - 120 x z + \tfrac{52238 y^{3}}{375} - \tfrac{75524 y^{2} z}{375} - \tfrac{357 y^{2}}{500} + \tfrac{30131 y z}{750} + \tfrac{8119 y}{1500}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(- 1252952 x^{2} + 1540952 x y + 662096 x z + 52333 x - 86096 y^{2} - 784192 y z - 30023 y + 180000 z^{2} - 119738 z + 8119\right)}{1500}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{496762 x^{3}}{375} + \tfrac{394162 x^{2} y}{75} + \tfrac{1654048 x^{2} z}{375} - \tfrac{223541 x^{2}}{60} + \tfrac{2379334 x y^{2}}{375} + \tfrac{4326668 x y z}{375} - \tfrac{7296871 x y}{750} + \tfrac{345562 x z^{2}}{75} - \tfrac{12164027 x z}{1500} + \tfrac{2592203 x}{750} + \tfrac{301762 y^{3}}{125} + \tfrac{520124 y^{2} z}{75} - \tfrac{2905739 y^{2}}{500} + \tfrac{755286 y z^{2}}{125} - \tfrac{3162163 y z}{300} + \tfrac{1113167 y}{250} + \tfrac{570524 z^{3}}{375} - \tfrac{1035917 z^{2}}{250} + \tfrac{1103267 z}{300} - \tfrac{527643}{500}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{z \left(- 262952 x z + 32869 x - 237904 y z + 29738 y - 334952 z^{2} + 250821 z - 26119\right)}{1500}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(442952 x y - 705904 x z + 32869 x - 122096 y^{2} - 176856 y z - 35214 y - 273904 z^{2} + 315773 z - 26119\right)}{1500}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(1162952 x^{2} - 842096 x y - 1486952 x z - 32083 x + 604192 y z + 29738 y - 273904 z^{2} + 315773 z - 26119\right)}{1500}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{69262 x^{3}}{375} - \tfrac{529048 x^{2} y}{375} - \tfrac{67462 x^{2} z}{75} + \tfrac{47051 x^{2}}{60} - \tfrac{170062 x y^{2}}{75} - \tfrac{1644668 x y z}{375} + \tfrac{4925777 x y}{1500} - \tfrac{574834 x z^{2}}{375} + \tfrac{1905871 x z}{750} - \tfrac{382039 x}{375} - \tfrac{390524 y^{3}}{375} - \tfrac{435786 y^{2} z}{125} + \tfrac{624917 y^{2}}{250} - \tfrac{244724 y z^{2}}{75} + \tfrac{1539013 y z}{300} - \tfrac{563267 y}{300} - \tfrac{102262 z^{3}}{125} + \tfrac{1022489 z^{2}}{500} - \tfrac{205771 z}{125} + \tfrac{209643}{500}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{45}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{3}\)
where \(f_{3}\) is the 3th face;
and \(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3.

\(\displaystyle \boldsymbol{\phi}_{45} = \left(\begin{array}{c}\displaystyle \begin{cases} \tfrac{z \left(2744992 x^{2} + 3374992 x y + 3883488 x z - 2966054 x - 1867496 y z - 188437 y - 2069992 z^{2} + 879747 z + 189562\right)}{3000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(2744992 x^{2} + 3838496 x y + 3419984 x z - 2966054 x - 3307496 y^{2} - 629992 y z + 1040059 y - 348749 z + 189562\right)}{3000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(1768496 x^{2} + 1507496 x y + 2789992 x z - 1737558 x - 188437 y - 348749 z + 189562\right)}{3000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{35062 x^{3}}{125} - \tfrac{342559 x^{2} y}{375} - \tfrac{38699 x^{2} z}{75} + \tfrac{282089 x^{2}}{500} - \tfrac{58162 x y^{2}}{75} - \tfrac{277868 x y z}{375} + \tfrac{3154459 x y}{3000} + \tfrac{4314 x z^{2}}{125} + \tfrac{176951 x z}{600} - \tfrac{34546 x}{125} - \tfrac{53437 y^{3}}{375} - \tfrac{5623 y^{2} z}{375} + \tfrac{33277 y^{2}}{120} + \tfrac{29813 y z^{2}}{75} - \tfrac{325423 y z}{1500} - \tfrac{382487 y}{3000} + \tfrac{101251 z^{3}}{375} - \tfrac{494257 z^{2}}{1000} + \tfrac{138941 z}{600} - \tfrac{3657}{500}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} \tfrac{3 z \left(- 1840 x y + 760 x z + 135 x - 640 y^{2} + 592 y z + 636 y + 360 z^{2} - 483 z - 24\right)}{8}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{3 y \left(- 1080 x y + 135 x + 32 y^{2} + 280 y z + 188 y - 35 z - 24\right)}{8}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{3 x \left(- 520 x^{2} + 112 x y + 1640 x z - 313 x - 640 y^{2} - 1360 y z + 636 y - 35 z - 24\right)}{8}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\765 x^{3} + 3432 x^{2} y + 2625 x^{2} z - \tfrac{17829 x^{2}}{8} + 4119 x y^{2} + 7344 x y z - \tfrac{49833 x y}{8} + 2955 x z^{2} - \tfrac{10107 x z}{2} + \tfrac{16965 x}{8} + 1452 y^{3} + 4269 y^{2} z - \tfrac{7101 y^{2}}{2} + 3912 y z^{2} - \tfrac{53163 y z}{8} + \tfrac{5511 y}{2} + 1095 z^{3} - \tfrac{22599 z^{2}}{8} + \tfrac{19095 z}{8} - 657&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} - 120 x^{2} y - 120 x y^{2} + 120 x y - \tfrac{213749 x z^{2}}{375} + \tfrac{168749 x z}{3000} - \tfrac{120937 y z^{2}}{375} + \tfrac{75937 y z}{3000} - \tfrac{189562 z^{3}}{375} + \tfrac{206343 z^{2}}{500} - \tfrac{76781 z}{1500}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\tfrac{y \left(- 360000 x^{2} + 989992 x y - 3059984 x z + 528749 x + 3487496 y^{2} - 3946496 y z - 1080559 y - 2024992 z^{2} + 2394554 z - 153562\right)}{3000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\tfrac{x \left(4229992 x^{2} + 247496 x y - 5431488 x z - 987747 x - 360000 y^{2} - 1574992 y z + 435937 y - 2024992 z^{2} + 2394554 z - 153562\right)}{3000}&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\- \tfrac{281251 x^{3}}{375} - \tfrac{205313 x^{2} y}{75} - \tfrac{323814 x^{2} z}{125} + \tfrac{2138257 x^{2}}{1000} - \tfrac{1119377 x y^{2}}{375} - \tfrac{2404132 x y z}{375} + \tfrac{7563673 x y}{1500} - \tfrac{236701 x z^{2}}{75} + \tfrac{3069349 x z}{600} - \tfrac{1218941 x}{600} - \tfrac{374063 y^{3}}{375} - \tfrac{268538 y^{2} z}{75} + \tfrac{319703 y^{2}}{120} - \tfrac{1461941 y z^{2}}{375} + \tfrac{18409541 y z}{3000} - \tfrac{6930013 y}{3000} - \tfrac{164438 z^{3}}{125} + \tfrac{1601161 z^{2}}{500} - \tfrac{632533 z}{250} + \tfrac{321657}{500}&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{46}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{4},\tfrac{1}{4},\tfrac{1}{4})\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{46} = \left(\begin{array}{c}\displaystyle \begin{cases} 4 z \left(8 z - 1\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\4 y \left(8 y - 1\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\4 x \left(8 x - 1\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\32 x^{2} + 64 x y + 64 x z - 60 x + 32 y^{2} + 64 y z - 60 y + 32 z^{2} - 60 z + 28&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{47}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{4},\tfrac{1}{4},\tfrac{1}{4})\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{47} = \left(\begin{array}{c}\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 4 z \left(8 z - 1\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\4 y \left(8 y - 1\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\4 x \left(8 x - 1\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\32 x^{2} + 64 x y + 64 x z - 60 x + 32 y^{2} + 64 y z - 60 y + 32 z^{2} - 60 z + 28&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{48}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{4},\tfrac{1}{4},\tfrac{1}{4})\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{48} = \left(\begin{array}{c}\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\0&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\\\displaystyle \begin{cases} 4 z \left(8 z - 1\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\4 y \left(8 y - 1\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\4 x \left(8 x - 1\right)&\text{in }\operatorname{Triangle}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\32 x^{2} + 64 x y + 64 x z - 60 x + 32 y^{2} + 64 y z - 60 y + 32 z^{2} - 60 z + 28&\text{in }\operatorname{Triangle}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.

References

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Element added01 August 2021
Element last updated01 August 2021