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Trimmed serendipity H(div)

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Exterior calculus names\(\mathcal{S}^-_{k}\Lambda^{d-1}(\square_d)\)
Cockburn–fu names\(\left[S_{2,k}^\square\right]_{d-1}\)
Orders\(1\leqslant k\)
Reference elementsquadrilateral, hexahedron
Polynomial set\(\mathcal{P}_{k-1}^d \oplus \mathcal{Z}^{(21)}_{k} \oplus \mathcal{Z}^{(22)}_{k}\) (quadrilateral)
\(\mathcal{P}_{k-1}^d \oplus \mathcal{Z}^{(21)}_{k} \oplus \mathcal{Z}^{(23)}_{k} \oplus \mathcal{Z}^{(24)}_{k} \oplus \mathcal{Z}^{(25)}_{k}\) (hexahedron)
↓ Show polynomial set definitions ↓
DOFsOn each facet: normal integral moments with an order \(k-1\) dPc space
On the interior of the reference element: integral moments with an order \(k-3\) vector dPc space, and integral moments with \(\left\{\nabla(p)\middle|p\text{ is an order \(k-1\) monomial}\right\}\)
CategoriesVector-valued elements, H(div) conforming elements

Implementations

Symfem"TSdiv"
↓ Show Symfem examples ↓

Examples

quadrilateral
order 1
quadrilateral
order 2
quadrilateral
order 3
hexahedron
order 1
hexahedron
order 2
hexahedron
order 3
  • \(R\) is the reference quadrilateral. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle - y\end{array}\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{3}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot\hat{\boldsymbol{n}}_{0}\)
where \(e_{0}\) is the 0th edge;
and \(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0.

\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 1 - y\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle x - 1\\\displaystyle 0\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle - x\\\displaystyle 0\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)\)

This DOF is associated with edge 3 of the reference element.
  • \(R\) is the reference quadrilateral. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y\\\displaystyle y^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle x y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 2 x y\\\displaystyle - y^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle - x^{2}\\\displaystyle 2 x y\end{array}\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{9}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{n}}_{0}\)
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0}\) is a parametrisation of \(e_{0}\).

\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 6 x y - 6 x + 3 y^{2} - 7 y + 4\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - 6 x y + 6 x + 3 y^{2} - y - 2\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle - 3 x^{2} - 6 x y + 7 x + 6 y - 4\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{1}\)
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0}\) is a parametrisation of \(e_{1}\).

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle - 3 x^{2} + 6 x y + x - 6 y + 2\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{n}}_{2}\)
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0}\) is a parametrisation of \(e_{2}\).

\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle x \left(- 3 x + 6 y - 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{2}\)
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0}\) is a parametrisation of \(e_{2}\).

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle x \left(- 3 x - 6 y + 5\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{n}}_{3}\)
where \(e_{3}\) is the 3th edge;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \(s_{0}\) is a parametrisation of \(e_{3}\).

\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y \left(- 6 x + 3 y + 1\right)\end{array}\right)\)

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{3}\)
where \(e_{3}\) is the 3th edge;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \(s_{0}\) is a parametrisation of \(e_{3}\).

\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y \left(6 x + 3 y - 5\right)\end{array}\right)\)

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{8}:\mathbf{v}\mapsto\displaystyle\int_{R}\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(R\) is the reference element.

\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 6 x \left(1 - x\right)\\\displaystyle 0\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 6 y \left(1 - y\right)\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
  • \(R\) is the reference quadrilateral. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y^{2}\\\displaystyle y^{3}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2} y\\\displaystyle x y^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{3}\\\displaystyle x^{2} y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 3 x y^{2}\\\displaystyle - y^{3}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle - x^{3}\\\displaystyle 3 x^{2} y\end{array}\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{16}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(2 s_{0}^{2} - 3 s_{0} + 1)\hat{\boldsymbol{n}}_{0}\)
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0}\) is a parametrisation of \(e_{0}\).

\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle \frac{9 x \left(- 2 x y + x + 2 y - 1\right)}{2}\\\displaystyle - 30 x^{2} y + 30 x^{2} - 9 x y^{2} + 45 x y - 36 x - 10 y^{3} + \frac{45 y^{2}}{2} - \frac{43 y}{2} + 9\end{array}\right)\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(4 s_{0} \left(1 - s_{0}\right))\hat{\boldsymbol{n}}_{0}\)
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0}\) is a parametrisation of \(e_{0}\).

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 15 x^{2} y - 15 x^{2} - 15 x y + 15 x - 10 y^{3} + 18 y^{2} - \frac{13 y}{2} - \frac{3}{2}\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle \frac{9 x \left(2 x y - x - 2 y + 1\right)}{2}\\\displaystyle - 30 x^{2} y + 30 x^{2} + 9 x y^{2} + 15 x y - 24 x - 10 y^{3} + \frac{27 y^{2}}{2} - \frac{13 y}{2} + 3\end{array}\right)\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(2 s_{0}^{2} - 3 s_{0} + 1)\hat{\boldsymbol{n}}_{1}\)
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0}\) is a parametrisation of \(e_{1}\).

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 10 x^{3} + 9 x^{2} y - \frac{45 x^{2}}{2} + 30 x y^{2} - 45 x y + \frac{43 x}{2} - 30 y^{2} + 36 y - 9\\\displaystyle \frac{9 y \left(2 x y - 2 x - y + 1\right)}{2}\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(4 s_{0} \left(1 - s_{0}\right))\hat{\boldsymbol{n}}_{1}\)
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0}\) is a parametrisation of \(e_{1}\).

\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 10 x^{3} - 18 x^{2} - 15 x y^{2} + 15 x y + \frac{13 x}{2} + 15 y^{2} - 15 y + \frac{3}{2}\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(s_{0} \left(2 s_{0} - 1\right))\hat{\boldsymbol{n}}_{1}\)
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0}\) is a parametrisation of \(e_{1}\).

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 10 x^{3} - 9 x^{2} y - \frac{27 x^{2}}{2} + 30 x y^{2} - 15 x y + \frac{13 x}{2} - 30 y^{2} + 24 y - 3\\\displaystyle \frac{9 y \left(- 2 x y + 2 x + y - 1\right)}{2}\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(2 s_{0}^{2} - 3 s_{0} + 1)\hat{\boldsymbol{n}}_{2}\)
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0}\) is a parametrisation of \(e_{2}\).

\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle \frac{x \left(- 20 x^{2} + 18 x y + 15 x - 60 y^{2} + 54 y - 13\right)}{2}\\\displaystyle \frac{9 y \left(2 x y - 2 x - y + 1\right)}{2}\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(4 s_{0} \left(1 - s_{0}\right))\hat{\boldsymbol{n}}_{2}\)
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0}\) is a parametrisation of \(e_{2}\).

\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle \frac{x \left(- 20 x^{2} + 24 x + 30 y^{2} - 30 y - 1\right)}{2}\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(s_{0} \left(2 s_{0} - 1\right))\hat{\boldsymbol{n}}_{2}\)
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0}\) is a parametrisation of \(e_{2}\).

\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle \frac{x \left(- 20 x^{2} - 18 x y + 33 x - 60 y^{2} + 66 y - 19\right)}{2}\\\displaystyle \frac{9 y \left(- 2 x y + 2 x + y - 1\right)}{2}\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{9}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(2 s_{0}^{2} - 3 s_{0} + 1)\hat{\boldsymbol{n}}_{3}\)
where \(e_{3}\) is the 3th edge;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \(s_{0}\) is a parametrisation of \(e_{3}\).

\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle \frac{9 x \left(- 2 x y + x + 2 y - 1\right)}{2}\\\displaystyle \frac{y \left(60 x^{2} - 18 x y - 54 x + 20 y^{2} - 15 y + 13\right)}{2}\end{array}\right)\)

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{10}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(4 s_{0} \left(1 - s_{0}\right))\hat{\boldsymbol{n}}_{3}\)
where \(e_{3}\) is the 3th edge;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \(s_{0}\) is a parametrisation of \(e_{3}\).

\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{y \left(- 30 x^{2} + 30 x + 20 y^{2} - 24 y + 1\right)}{2}\end{array}\right)\)

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{11}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(s_{0} \left(2 s_{0} - 1\right))\hat{\boldsymbol{n}}_{3}\)
where \(e_{3}\) is the 3th edge;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \(s_{0}\) is a parametrisation of \(e_{3}\).

\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle \frac{9 x \left(2 x y - x - 2 y + 1\right)}{2}\\\displaystyle \frac{y \left(60 x^{2} + 18 x y - 66 x + 20 y^{2} - 33 y + 19\right)}{2}\end{array}\right)\)

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{12}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)
where \(R\) is the reference element.

\(\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 3 x \left(20 x^{2} + 6 x y - 35 x - 6 y + 15\right)\\\displaystyle 9 y \left(2 x y - 2 x - y + 1\right)\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)
where \(R\) is the reference element.

\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 9 x \left(2 x y - x - 2 y + 1\right)\\\displaystyle 3 y \left(6 x y - 6 x + 20 y^{2} - 35 y + 15\right)\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{14}:\mathbf{v}\mapsto\displaystyle\int_{R}\left(\begin{array}{c}\displaystyle 2 s_{0}\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).

\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 30 x \left(- 2 x^{2} + 3 x - 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{15}:\mathbf{v}\mapsto\displaystyle\int_{R}\left(\begin{array}{c}\displaystyle s_{1}\\\displaystyle s_{0}\end{array}\right)\cdot\mathbf{v}\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).

\(\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 18 x \left(- 2 x y + x + 2 y - 1\right)\\\displaystyle 18 y \left(- 2 x y + 2 x + y - 1\right)\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{16}:\mathbf{v}\mapsto\displaystyle\int_{R}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 s_{1}\end{array}\right)\cdot\mathbf{v}\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).

\(\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 30 y \left(- 2 y^{2} + 3 y - 1\right)\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
  • \(R\) is the reference hexahedron. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle y\\\displaystyle z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 2 x\\\displaystyle - y\\\displaystyle - z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle - 2 y\\\displaystyle z\end{array}\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{5}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot\hat{\boldsymbol{n}}_{0}\)
where \(f_{0}\) is the 0th face;
and \(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0.

\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1 - z\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y - 1\\\displaystyle 0\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 1 - x\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot\hat{\boldsymbol{n}}_{4}\)
where \(f_{4}\) is the 4th face;
and \(\hat{\boldsymbol{n}}_{4}\) is the normal to facet 4.

\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - y\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 4 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot\hat{\boldsymbol{n}}_{5}\)
where \(f_{5}\) is the 5th face;
and \(\hat{\boldsymbol{n}}_{5}\) is the normal to facet 5.

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)\)

This DOF is associated with face 5 of the reference element.
  • \(R\) is the reference hexahedron. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x z\\\displaystyle y z\\\displaystyle z^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y\\\displaystyle y^{2}\\\displaystyle y z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle x y\\\displaystyle x z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 3 x z\\\displaystyle - y z\\\displaystyle - z^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x z\\\displaystyle - 3 y z\\\displaystyle z^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle - x y\\\displaystyle - y^{2}\\\displaystyle 3 y z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 3 x y\\\displaystyle - y^{2}\\\displaystyle - y z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle - 3 x y\\\displaystyle x z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle - x^{2}\\\displaystyle - x y\\\displaystyle 3 x z\end{array}\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{20}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(- s_{0} - s_{1} + 1)\hat{\boldsymbol{n}}_{0}\)
where \(f_{0}\) is the 0th face;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{0}\).

\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 6 x z - 6 x + 6 y z - 6 y + 3 z^{2} - 10 z + 7\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - 6 x z + 6 x + 6 y z - 6 y + 3 z^{2} - 4 z + 1\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 6 x z - 6 x - 6 y z + 6 y + 3 z^{2} - 4 z + 1\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - 6 x y + 6 x - 3 y^{2} - 6 y z + 10 y + 6 z - 7\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{1}\)
where \(f_{1}\) is the 1st face;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{1}\).

\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 6 x y - 6 x - 3 y^{2} - 6 y z + 4 y + 6 z - 1\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(s_{1})\hat{\boldsymbol{n}}_{1}\)
where \(f_{1}\) is the 1st face;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{1}\).

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - 6 x y + 6 x - 3 y^{2} + 6 y z + 4 y - 6 z - 1\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(- s_{0} - s_{1} + 1)\hat{\boldsymbol{n}}_{2}\)
where \(f_{2}\) is the 2nd face;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{2}\).

\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 3 x^{2} + 6 x y + 6 x z - 10 x - 6 y - 6 z + 7\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{2}\)
where \(f_{2}\) is the 2nd face;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{2}\).

\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 3 x^{2} - 6 x y + 6 x z - 4 x + 6 y - 6 z + 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(s_{1})\hat{\boldsymbol{n}}_{2}\)
where \(f_{2}\) is the 2nd face;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{2}\).

\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 3 x^{2} + 6 x y - 6 x z - 4 x - 6 y + 6 z + 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{9}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(- s_{0} - s_{1} + 1)\hat{\boldsymbol{n}}_{3}\)
where \(f_{3}\) is the 3th face;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{3}\).

\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle x \left(3 x - 6 y - 6 z + 4\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{10}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{3}\)
where \(f_{3}\) is the 3th face;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{3}\).

\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle x \left(3 x + 6 y - 6 z - 2\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{11}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(s_{1})\hat{\boldsymbol{n}}_{3}\)
where \(f_{3}\) is the 3th face;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{3}\).

\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle x \left(3 x - 6 y + 6 z - 2\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{12}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot(- s_{0} - s_{1} + 1)\hat{\boldsymbol{n}}_{4}\)
where \(f_{4}\) is the 4th face;
\(\hat{\boldsymbol{n}}_{4}\) is the normal to facet 4;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{4}\).

\(\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y \left(6 x - 3 y + 6 z - 4\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 4 of the reference element.
\(\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{4}\)
where \(f_{4}\) is the 4th face;
\(\hat{\boldsymbol{n}}_{4}\) is the normal to facet 4;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{4}\).

\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y \left(- 6 x - 3 y + 6 z + 2\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 4 of the reference element.
\(\displaystyle l_{14}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot(s_{1})\hat{\boldsymbol{n}}_{4}\)
where \(f_{4}\) is the 4th face;
\(\hat{\boldsymbol{n}}_{4}\) is the normal to facet 4;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{4}\).

\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y \left(6 x - 3 y - 6 z + 2\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 4 of the reference element.
\(\displaystyle l_{15}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot(- s_{0} - s_{1} + 1)\hat{\boldsymbol{n}}_{5}\)
where \(f_{5}\) is the 5th face;
\(\hat{\boldsymbol{n}}_{5}\) is the normal to facet 5;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{5}\).

\(\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z \left(- 6 x - 6 y + 3 z + 4\right)\end{array}\right)\)

This DOF is associated with face 5 of the reference element.
\(\displaystyle l_{16}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{5}\)
where \(f_{5}\) is the 5th face;
\(\hat{\boldsymbol{n}}_{5}\) is the normal to facet 5;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{5}\).

\(\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z \left(6 x - 6 y + 3 z - 2\right)\end{array}\right)\)

This DOF is associated with face 5 of the reference element.
\(\displaystyle l_{17}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot(s_{1})\hat{\boldsymbol{n}}_{5}\)
where \(f_{5}\) is the 5th face;
\(\hat{\boldsymbol{n}}_{5}\) is the normal to facet 5;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{5}\).

\(\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z \left(- 6 x + 6 y + 3 z - 2\right)\end{array}\right)\)

This DOF is associated with face 5 of the reference element.
\(\displaystyle l_{18}:\mathbf{v}\mapsto\displaystyle\int_{R}\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(R\) is the reference element.

\(\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle 6 x \left(1 - x\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{19}:\mathbf{v}\mapsto\displaystyle\int_{R}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)\cdot\mathbf{v}\)
where \(R\) is the reference element.

\(\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 6 z \left(1 - z\right)\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{20}:\mathbf{v}\mapsto\displaystyle\int_{R}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(R\) is the reference element.

\(\displaystyle \boldsymbol{\phi}_{20} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 6 y \left(1 - y\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
  • \(R\) is the reference hexahedron. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle z^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle z^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x z^{2}\\\displaystyle y z^{2}\\\displaystyle z^{3}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y z\\\displaystyle y^{2} z\\\displaystyle y z^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y^{2}\\\displaystyle y^{3}\\\displaystyle y^{2} z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2} z\\\displaystyle x y z\\\displaystyle x z^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2} y\\\displaystyle x y^{2}\\\displaystyle x y z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{3}\\\displaystyle x^{2} y\\\displaystyle x^{2} z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 4 x z^{2}\\\displaystyle - y z^{2}\\\displaystyle - z^{3}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x z^{2}\\\displaystyle - 4 y z^{2}\\\displaystyle z^{3}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle - x y^{2}\\\displaystyle - y^{3}\\\displaystyle 4 y^{2} z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 4 x y z\\\displaystyle - y^{2} z\\\displaystyle - y z^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2} z\\\displaystyle - 4 x y z\\\displaystyle x z^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle - x^{2} y\\\displaystyle - x y^{2}\\\displaystyle 4 x y z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 4 x y^{2}\\\displaystyle - y^{3}\\\displaystyle - y^{2} z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{3}\\\displaystyle - 4 x^{2} y\\\displaystyle x^{2} z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle - x^{3}\\\displaystyle - x^{2} y\\\displaystyle 4 x^{2} z\end{array}\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{44}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(2 s_{0}^{2} + 4 s_{0} s_{1} - 3 s_{0} + 2 s_{1}^{2} - 3 s_{1} + 1)\hat{\boldsymbol{n}}_{0}\)
where \(f_{0}\) is the 0th face;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{0}\).

\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle \frac{9 x \left(- 2 x z + x + 2 z - 1\right)}{2}\\\displaystyle \frac{9 y \left(- 2 y z + y + 2 z - 1\right)}{2}\\\displaystyle - 30 x^{2} z + 30 x^{2} - 36 x y z + 36 x y - 9 x z^{2} + 63 x z - 54 x - 30 y^{2} z + 30 y^{2} - 9 y z^{2} + 63 y z - 54 y - 10 z^{3} + 27 z^{2} - 43 z + 26\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(4 s_{0} \left(- s_{0} - s_{1} + 1\right))\hat{\boldsymbol{n}}_{0}\)
where \(f_{0}\) is the 0th face;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{0}\).

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{9 y \left(- 2 y z + y + 2 z - 1\right)}{2}\\\displaystyle 15 x^{2} z - 15 x^{2} - 15 x z + 15 x - 30 y^{2} z + 30 y^{2} - 9 y z^{2} + 45 y z - 36 y - 10 z^{3} + \frac{45 z^{2}}{2} - 19 z + \frac{13}{2}\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(s_{0} \left(2 s_{0} - 1\right))\hat{\boldsymbol{n}}_{0}\)
where \(f_{0}\) is the 0th face;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{0}\).

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle \frac{9 x \left(2 x z - x - 2 z + 1\right)}{2}\\\displaystyle \frac{9 y \left(- 2 y z + y + 2 z - 1\right)}{2}\\\displaystyle - 30 x^{2} z + 30 x^{2} + 36 x y z - 36 x y + 9 x z^{2} - 3 x z - 6 x - 30 y^{2} z + 30 y^{2} - 9 y z^{2} + 27 y z - 18 y - 10 z^{3} + 18 z^{2} - 10 z + 2\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(4 s_{1} \left(- s_{0} - s_{1} + 1\right))\hat{\boldsymbol{n}}_{0}\)
where \(f_{0}\) is the 0th face;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{0}\).

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle \frac{9 x \left(- 2 x z + x + 2 z - 1\right)}{2}\\\displaystyle 0\\\displaystyle - 30 x^{2} z + 30 x^{2} - 9 x z^{2} + 45 x z - 36 x + 15 y^{2} z - 15 y^{2} - 15 y z + 15 y - 10 z^{3} + \frac{45 z^{2}}{2} - 19 z + \frac{13}{2}\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(4 s_{0} s_{1})\hat{\boldsymbol{n}}_{0}\)
where \(f_{0}\) is the 0th face;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{0}\).

\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 15 x^{2} z - 15 x^{2} - 15 x z + 15 x + 15 y^{2} z - 15 y^{2} - 15 y z + 15 y - 10 z^{3} + 18 z^{2} - 4 z - 4\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(s_{1} \left(2 s_{1} - 1\right))\hat{\boldsymbol{n}}_{0}\)
where \(f_{0}\) is the 0th face;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{0}\).

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle \frac{9 x \left(- 2 x z + x + 2 z - 1\right)}{2}\\\displaystyle \frac{9 y \left(2 y z - y - 2 z + 1\right)}{2}\\\displaystyle - 30 x^{2} z + 30 x^{2} + 36 x y z - 36 x y - 9 x z^{2} + 27 x z - 18 x - 30 y^{2} z + 30 y^{2} + 9 y z^{2} - 3 y z - 6 y - 10 z^{3} + 18 z^{2} - 10 z + 2\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(2 s_{0}^{2} + 4 s_{0} s_{1} - 3 s_{0} + 2 s_{1}^{2} - 3 s_{1} + 1)\hat{\boldsymbol{n}}_{1}\)
where \(f_{1}\) is the 1st face;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{1}\).

\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle \frac{9 x \left(2 x y - x - 2 y + 1\right)}{2}\\\displaystyle 30 x^{2} y - 30 x^{2} + 9 x y^{2} + 36 x y z - 63 x y - 36 x z + 54 x + 10 y^{3} + 9 y^{2} z - 27 y^{2} + 30 y z^{2} - 63 y z + 43 y - 30 z^{2} + 54 z - 26\\\displaystyle \frac{9 z \left(2 y z - 2 y - z + 1\right)}{2}\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(4 s_{0} \left(- s_{0} - s_{1} + 1\right))\hat{\boldsymbol{n}}_{1}\)
where \(f_{1}\) is the 1st face;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{1}\).

\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - 15 x^{2} y + 15 x^{2} + 15 x y - 15 x + 10 y^{3} + 9 y^{2} z - \frac{45 y^{2}}{2} + 30 y z^{2} - 45 y z + 19 y - 30 z^{2} + 36 z - \frac{13}{2}\\\displaystyle \frac{9 z \left(2 y z - 2 y - z + 1\right)}{2}\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(s_{0} \left(2 s_{0} - 1\right))\hat{\boldsymbol{n}}_{1}\)
where \(f_{1}\) is the 1st face;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{1}\).

\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle \frac{9 x \left(- 2 x y + x + 2 y - 1\right)}{2}\\\displaystyle 30 x^{2} y - 30 x^{2} - 9 x y^{2} - 36 x y z + 3 x y + 36 x z + 6 x + 10 y^{3} + 9 y^{2} z - 18 y^{2} + 30 y z^{2} - 27 y z + 10 y - 30 z^{2} + 18 z - 2\\\displaystyle \frac{9 z \left(2 y z - 2 y - z + 1\right)}{2}\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{9}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(4 s_{1} \left(- s_{0} - s_{1} + 1\right))\hat{\boldsymbol{n}}_{1}\)
where \(f_{1}\) is the 1st face;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{1}\).

\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle \frac{9 x \left(2 x y - x - 2 y + 1\right)}{2}\\\displaystyle 30 x^{2} y - 30 x^{2} + 9 x y^{2} - 45 x y + 36 x + 10 y^{3} - \frac{45 y^{2}}{2} - 15 y z^{2} + 15 y z + 19 y + 15 z^{2} - 15 z - \frac{13}{2}\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{10}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(4 s_{0} s_{1})\hat{\boldsymbol{n}}_{1}\)
where \(f_{1}\) is the 1st face;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{1}\).

\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - 15 x^{2} y + 15 x^{2} + 15 x y - 15 x + 10 y^{3} - 18 y^{2} - 15 y z^{2} + 15 y z + 4 y + 15 z^{2} - 15 z + 4\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{11}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(s_{1} \left(2 s_{1} - 1\right))\hat{\boldsymbol{n}}_{1}\)
where \(f_{1}\) is the 1st face;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{1}\).

\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle \frac{9 x \left(2 x y - x - 2 y + 1\right)}{2}\\\displaystyle 30 x^{2} y - 30 x^{2} + 9 x y^{2} - 36 x y z - 27 x y + 36 x z + 18 x + 10 y^{3} - 9 y^{2} z - 18 y^{2} + 30 y z^{2} + 3 y z + 10 y - 30 z^{2} + 6 z - 2\\\displaystyle \frac{9 z \left(- 2 y z + 2 y + z - 1\right)}{2}\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{12}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(2 s_{0}^{2} + 4 s_{0} s_{1} - 3 s_{0} + 2 s_{1}^{2} - 3 s_{1} + 1)\hat{\boldsymbol{n}}_{2}\)
where \(f_{2}\) is the 2nd face;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{2}\).

\(\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle - 10 x^{3} - 9 x^{2} y - 9 x^{2} z + 27 x^{2} - 30 x y^{2} - 36 x y z + 63 x y - 30 x z^{2} + 63 x z - 43 x + 30 y^{2} + 36 y z - 54 y + 30 z^{2} - 54 z + 26\\\displaystyle \frac{9 y \left(- 2 x y + 2 x + y - 1\right)}{2}\\\displaystyle \frac{9 z \left(- 2 x z + 2 x + z - 1\right)}{2}\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(4 s_{0} \left(- s_{0} - s_{1} + 1\right))\hat{\boldsymbol{n}}_{2}\)
where \(f_{2}\) is the 2nd face;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{2}\).

\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle - 10 x^{3} - 9 x^{2} z + \frac{45 x^{2}}{2} + 15 x y^{2} - 15 x y - 30 x z^{2} + 45 x z - 19 x - 15 y^{2} + 15 y + 30 z^{2} - 36 z + \frac{13}{2}\\\displaystyle 0\\\displaystyle \frac{9 z \left(- 2 x z + 2 x + z - 1\right)}{2}\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{14}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(s_{0} \left(2 s_{0} - 1\right))\hat{\boldsymbol{n}}_{2}\)
where \(f_{2}\) is the 2nd face;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{2}\).

\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle - 10 x^{3} + 9 x^{2} y - 9 x^{2} z + 18 x^{2} - 30 x y^{2} + 36 x y z - 3 x y - 30 x z^{2} + 27 x z - 10 x + 30 y^{2} - 36 y z - 6 y + 30 z^{2} - 18 z + 2\\\displaystyle \frac{9 y \left(2 x y - 2 x - y + 1\right)}{2}\\\displaystyle \frac{9 z \left(- 2 x z + 2 x + z - 1\right)}{2}\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{15}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(4 s_{1} \left(- s_{0} - s_{1} + 1\right))\hat{\boldsymbol{n}}_{2}\)
where \(f_{2}\) is the 2nd face;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{2}\).

\(\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle - 10 x^{3} - 9 x^{2} y + \frac{45 x^{2}}{2} - 30 x y^{2} + 45 x y + 15 x z^{2} - 15 x z - 19 x + 30 y^{2} - 36 y - 15 z^{2} + 15 z + \frac{13}{2}\\\displaystyle \frac{9 y \left(- 2 x y + 2 x + y - 1\right)}{2}\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{16}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(4 s_{0} s_{1})\hat{\boldsymbol{n}}_{2}\)
where \(f_{2}\) is the 2nd face;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{2}\).

\(\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle - 10 x^{3} + 18 x^{2} + 15 x y^{2} - 15 x y + 15 x z^{2} - 15 x z - 4 x - 15 y^{2} + 15 y - 15 z^{2} + 15 z - 4\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{17}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(s_{1} \left(2 s_{1} - 1\right))\hat{\boldsymbol{n}}_{2}\)
where \(f_{2}\) is the 2nd face;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{2}\).

\(\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle - 10 x^{3} - 9 x^{2} y + 9 x^{2} z + 18 x^{2} - 30 x y^{2} + 36 x y z + 27 x y - 30 x z^{2} - 3 x z - 10 x + 30 y^{2} - 36 y z - 18 y + 30 z^{2} - 6 z + 2\\\displaystyle \frac{9 y \left(- 2 x y + 2 x + y - 1\right)}{2}\\\displaystyle \frac{9 z \left(2 x z - 2 x - z + 1\right)}{2}\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{18}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(2 s_{0}^{2} + 4 s_{0} s_{1} - 3 s_{0} + 2 s_{1}^{2} - 3 s_{1} + 1)\hat{\boldsymbol{n}}_{3}\)
where \(f_{3}\) is the 3th face;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{3}\).

\(\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle x \left(10 x^{2} - 9 x y - 9 x z - 3 x + 30 y^{2} + 36 y z - 45 y + 30 z^{2} - 45 z + 19\right)\\\displaystyle \frac{9 y \left(- 2 x y + 2 x + y - 1\right)}{2}\\\displaystyle \frac{9 z \left(- 2 x z + 2 x + z - 1\right)}{2}\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{19}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(4 s_{0} \left(- s_{0} - s_{1} + 1\right))\hat{\boldsymbol{n}}_{3}\)
where \(f_{3}\) is the 3th face;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{3}\).

\(\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle \frac{x \left(20 x^{2} - 18 x z - 15 x - 30 y^{2} + 30 y + 60 z^{2} - 54 z + 8\right)}{2}\\\displaystyle 0\\\displaystyle \frac{9 z \left(- 2 x z + 2 x + z - 1\right)}{2}\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{20}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(s_{0} \left(2 s_{0} - 1\right))\hat{\boldsymbol{n}}_{3}\)
where \(f_{3}\) is the 3th face;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{3}\).

\(\displaystyle \boldsymbol{\phi}_{20} = \left(\begin{array}{c}\displaystyle x \left(10 x^{2} + 9 x y - 9 x z - 12 x + 30 y^{2} - 36 y z - 15 y + 30 z^{2} - 9 z + 4\right)\\\displaystyle \frac{9 y \left(2 x y - 2 x - y + 1\right)}{2}\\\displaystyle \frac{9 z \left(- 2 x z + 2 x + z - 1\right)}{2}\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{21}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(4 s_{1} \left(- s_{0} - s_{1} + 1\right))\hat{\boldsymbol{n}}_{3}\)
where \(f_{3}\) is the 3th face;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{3}\).

\(\displaystyle \boldsymbol{\phi}_{21} = \left(\begin{array}{c}\displaystyle \frac{x \left(20 x^{2} - 18 x y - 15 x + 60 y^{2} - 54 y - 30 z^{2} + 30 z + 8\right)}{2}\\\displaystyle \frac{9 y \left(- 2 x y + 2 x + y - 1\right)}{2}\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{22}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(4 s_{0} s_{1})\hat{\boldsymbol{n}}_{3}\)
where \(f_{3}\) is the 3th face;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{3}\).

\(\displaystyle \boldsymbol{\phi}_{22} = \left(\begin{array}{c}\displaystyle x \left(10 x^{2} - 12 x - 15 y^{2} + 15 y - 15 z^{2} + 15 z - 2\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{23}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(s_{1} \left(2 s_{1} - 1\right))\hat{\boldsymbol{n}}_{3}\)
where \(f_{3}\) is the 3th face;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{3}\).

\(\displaystyle \boldsymbol{\phi}_{23} = \left(\begin{array}{c}\displaystyle x \left(10 x^{2} - 9 x y + 9 x z - 12 x + 30 y^{2} - 36 y z - 9 y + 30 z^{2} - 15 z + 4\right)\\\displaystyle \frac{9 y \left(- 2 x y + 2 x + y - 1\right)}{2}\\\displaystyle \frac{9 z \left(2 x z - 2 x - z + 1\right)}{2}\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{24}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot(2 s_{0}^{2} + 4 s_{0} s_{1} - 3 s_{0} + 2 s_{1}^{2} - 3 s_{1} + 1)\hat{\boldsymbol{n}}_{4}\)
where \(f_{4}\) is the 4th face;
\(\hat{\boldsymbol{n}}_{4}\) is the normal to facet 4;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{4}\).

\(\displaystyle \boldsymbol{\phi}_{24} = \left(\begin{array}{c}\displaystyle \frac{9 x \left(2 x y - x - 2 y + 1\right)}{2}\\\displaystyle y \left(- 30 x^{2} + 9 x y - 36 x z + 45 x - 10 y^{2} + 9 y z + 3 y - 30 z^{2} + 45 z - 19\right)\\\displaystyle \frac{9 z \left(2 y z - 2 y - z + 1\right)}{2}\end{array}\right)\)

This DOF is associated with face 4 of the reference element.
\(\displaystyle l_{25}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot(4 s_{0} \left(- s_{0} - s_{1} + 1\right))\hat{\boldsymbol{n}}_{4}\)
where \(f_{4}\) is the 4th face;
\(\hat{\boldsymbol{n}}_{4}\) is the normal to facet 4;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{4}\).

\(\displaystyle \boldsymbol{\phi}_{25} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{y \left(30 x^{2} - 30 x - 20 y^{2} + 18 y z + 15 y - 60 z^{2} + 54 z - 8\right)}{2}\\\displaystyle \frac{9 z \left(2 y z - 2 y - z + 1\right)}{2}\end{array}\right)\)

This DOF is associated with face 4 of the reference element.
\(\displaystyle l_{26}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot(s_{0} \left(2 s_{0} - 1\right))\hat{\boldsymbol{n}}_{4}\)
where \(f_{4}\) is the 4th face;
\(\hat{\boldsymbol{n}}_{4}\) is the normal to facet 4;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{4}\).

\(\displaystyle \boldsymbol{\phi}_{26} = \left(\begin{array}{c}\displaystyle \frac{9 x \left(- 2 x y + x + 2 y - 1\right)}{2}\\\displaystyle y \left(- 30 x^{2} - 9 x y + 36 x z + 15 x - 10 y^{2} + 9 y z + 12 y - 30 z^{2} + 9 z - 4\right)\\\displaystyle \frac{9 z \left(2 y z - 2 y - z + 1\right)}{2}\end{array}\right)\)

This DOF is associated with face 4 of the reference element.
\(\displaystyle l_{27}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot(4 s_{1} \left(- s_{0} - s_{1} + 1\right))\hat{\boldsymbol{n}}_{4}\)
where \(f_{4}\) is the 4th face;
\(\hat{\boldsymbol{n}}_{4}\) is the normal to facet 4;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{4}\).

\(\displaystyle \boldsymbol{\phi}_{27} = \left(\begin{array}{c}\displaystyle \frac{9 x \left(2 x y - x - 2 y + 1\right)}{2}\\\displaystyle \frac{y \left(- 60 x^{2} + 18 x y + 54 x - 20 y^{2} + 15 y + 30 z^{2} - 30 z - 8\right)}{2}\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 4 of the reference element.
\(\displaystyle l_{28}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot(4 s_{0} s_{1})\hat{\boldsymbol{n}}_{4}\)
where \(f_{4}\) is the 4th face;
\(\hat{\boldsymbol{n}}_{4}\) is the normal to facet 4;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{4}\).

\(\displaystyle \boldsymbol{\phi}_{28} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y \left(15 x^{2} - 15 x - 10 y^{2} + 12 y + 15 z^{2} - 15 z + 2\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 4 of the reference element.
\(\displaystyle l_{29}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot(s_{1} \left(2 s_{1} - 1\right))\hat{\boldsymbol{n}}_{4}\)
where \(f_{4}\) is the 4th face;
\(\hat{\boldsymbol{n}}_{4}\) is the normal to facet 4;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{4}\).

\(\displaystyle \boldsymbol{\phi}_{29} = \left(\begin{array}{c}\displaystyle \frac{9 x \left(2 x y - x - 2 y + 1\right)}{2}\\\displaystyle y \left(- 30 x^{2} + 9 x y + 36 x z + 9 x - 10 y^{2} - 9 y z + 12 y - 30 z^{2} + 15 z - 4\right)\\\displaystyle \frac{9 z \left(- 2 y z + 2 y + z - 1\right)}{2}\end{array}\right)\)

This DOF is associated with face 4 of the reference element.
\(\displaystyle l_{30}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot(2 s_{0}^{2} + 4 s_{0} s_{1} - 3 s_{0} + 2 s_{1}^{2} - 3 s_{1} + 1)\hat{\boldsymbol{n}}_{5}\)
where \(f_{5}\) is the 5th face;
\(\hat{\boldsymbol{n}}_{5}\) is the normal to facet 5;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{5}\).

\(\displaystyle \boldsymbol{\phi}_{30} = \left(\begin{array}{c}\displaystyle \frac{9 x \left(- 2 x z + x + 2 z - 1\right)}{2}\\\displaystyle \frac{9 y \left(- 2 y z + y + 2 z - 1\right)}{2}\\\displaystyle z \left(30 x^{2} + 36 x y - 9 x z - 45 x + 30 y^{2} - 9 y z - 45 y + 10 z^{2} - 3 z + 19\right)\end{array}\right)\)

This DOF is associated with face 5 of the reference element.
\(\displaystyle l_{31}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot(4 s_{0} \left(- s_{0} - s_{1} + 1\right))\hat{\boldsymbol{n}}_{5}\)
where \(f_{5}\) is the 5th face;
\(\hat{\boldsymbol{n}}_{5}\) is the normal to facet 5;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{5}\).

\(\displaystyle \boldsymbol{\phi}_{31} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{9 y \left(- 2 y z + y + 2 z - 1\right)}{2}\\\displaystyle \frac{z \left(- 30 x^{2} + 30 x + 60 y^{2} - 18 y z - 54 y + 20 z^{2} - 15 z + 8\right)}{2}\end{array}\right)\)

This DOF is associated with face 5 of the reference element.
\(\displaystyle l_{32}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot(s_{0} \left(2 s_{0} - 1\right))\hat{\boldsymbol{n}}_{5}\)
where \(f_{5}\) is the 5th face;
\(\hat{\boldsymbol{n}}_{5}\) is the normal to facet 5;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{5}\).

\(\displaystyle \boldsymbol{\phi}_{32} = \left(\begin{array}{c}\displaystyle \frac{9 x \left(2 x z - x - 2 z + 1\right)}{2}\\\displaystyle \frac{9 y \left(- 2 y z + y + 2 z - 1\right)}{2}\\\displaystyle z \left(30 x^{2} - 36 x y + 9 x z - 15 x + 30 y^{2} - 9 y z - 9 y + 10 z^{2} - 12 z + 4\right)\end{array}\right)\)

This DOF is associated with face 5 of the reference element.
\(\displaystyle l_{33}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot(4 s_{1} \left(- s_{0} - s_{1} + 1\right))\hat{\boldsymbol{n}}_{5}\)
where \(f_{5}\) is the 5th face;
\(\hat{\boldsymbol{n}}_{5}\) is the normal to facet 5;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{5}\).

\(\displaystyle \boldsymbol{\phi}_{33} = \left(\begin{array}{c}\displaystyle \frac{9 x \left(- 2 x z + x + 2 z - 1\right)}{2}\\\displaystyle 0\\\displaystyle \frac{z \left(60 x^{2} - 18 x z - 54 x - 30 y^{2} + 30 y + 20 z^{2} - 15 z + 8\right)}{2}\end{array}\right)\)

This DOF is associated with face 5 of the reference element.
\(\displaystyle l_{34}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot(4 s_{0} s_{1})\hat{\boldsymbol{n}}_{5}\)
where \(f_{5}\) is the 5th face;
\(\hat{\boldsymbol{n}}_{5}\) is the normal to facet 5;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{5}\).

\(\displaystyle \boldsymbol{\phi}_{34} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z \left(- 15 x^{2} + 15 x - 15 y^{2} + 15 y + 10 z^{2} - 12 z - 2\right)\end{array}\right)\)

This DOF is associated with face 5 of the reference element.
\(\displaystyle l_{35}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot(s_{1} \left(2 s_{1} - 1\right))\hat{\boldsymbol{n}}_{5}\)
where \(f_{5}\) is the 5th face;
\(\hat{\boldsymbol{n}}_{5}\) is the normal to facet 5;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{5}\).

\(\displaystyle \boldsymbol{\phi}_{35} = \left(\begin{array}{c}\displaystyle \frac{9 x \left(- 2 x z + x + 2 z - 1\right)}{2}\\\displaystyle \frac{9 y \left(2 y z - y - 2 z + 1\right)}{2}\\\displaystyle z \left(30 x^{2} - 36 x y - 9 x z - 9 x + 30 y^{2} + 9 y z - 15 y + 10 z^{2} - 12 z + 4\right)\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{36} = \left(\begin{array}{c}\displaystyle 6 x \left(10 x^{2} + 3 x y + 3 x z - 19 x - 3 y - 3 z + 9\right)\\\displaystyle 9 y \left(2 x y - 2 x - y + 1\right)\\\displaystyle 9 z \left(2 x z - 2 x - z + 1\right)\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{37} = \left(\begin{array}{c}\displaystyle 9 x \left(2 x y - x - 2 y + 1\right)\\\displaystyle 6 y \left(3 x y - 3 x + 10 y^{2} + 3 y z - 19 y - 3 z + 9\right)\\\displaystyle 9 z \left(2 y z - 2 y - z + 1\right)\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{38} = \left(\begin{array}{c}\displaystyle 9 x \left(2 x z - x - 2 z + 1\right)\\\displaystyle 9 y \left(2 y z - y - 2 z + 1\right)\\\displaystyle 6 z \left(3 x z - 3 x + 3 y z - 3 y + 10 z^{2} - 19 z + 9\right)\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{39}:\mathbf{v}\mapsto\displaystyle\int_{R}\left(\begin{array}{c}\displaystyle 2 s_{0}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(R\) is the reference element;
and \((s_{0},s_{1},s_{2})\) is a parametrisation of \(R(3)\).

\(\displaystyle \boldsymbol{\phi}_{39} = \left(\begin{array}{c}\displaystyle 30 x \left(- 2 x^{2} + 3 x - 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{40}:\mathbf{v}\mapsto\displaystyle\int_{R}\left(\begin{array}{c}\displaystyle s_{2}\\\displaystyle 0\\\displaystyle s_{0}\end{array}\right)\cdot\mathbf{v}\)
where \(R\) is the reference element;
and \((s_{0},s_{1},s_{2})\) is a parametrisation of \(R(3)\).

\(\displaystyle \boldsymbol{\phi}_{40} = \left(\begin{array}{c}\displaystyle 18 x \left(- 2 x z + x + 2 z - 1\right)\\\displaystyle 0\\\displaystyle 18 z \left(- 2 x z + 2 x + z - 1\right)\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{41}:\mathbf{v}\mapsto\displaystyle\int_{R}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 2 s_{2}\end{array}\right)\cdot\mathbf{v}\)
where \(R\) is the reference element;
and \((s_{0},s_{1},s_{2})\) is a parametrisation of \(R(3)\).

\(\displaystyle \boldsymbol{\phi}_{41} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 30 z \left(- 2 z^{2} + 3 z - 1\right)\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{42}:\mathbf{v}\mapsto\displaystyle\int_{R}\left(\begin{array}{c}\displaystyle s_{1}\\\displaystyle s_{0}\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(R\) is the reference element;
and \((s_{0},s_{1},s_{2})\) is a parametrisation of \(R(3)\).

\(\displaystyle \boldsymbol{\phi}_{42} = \left(\begin{array}{c}\displaystyle 18 x \left(- 2 x y + x + 2 y - 1\right)\\\displaystyle 18 y \left(- 2 x y + 2 x + y - 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{43}:\mathbf{v}\mapsto\displaystyle\int_{R}\left(\begin{array}{c}\displaystyle 0\\\displaystyle s_{2}\\\displaystyle s_{1}\end{array}\right)\cdot\mathbf{v}\)
where \(R\) is the reference element;
and \((s_{0},s_{1},s_{2})\) is a parametrisation of \(R(3)\).

\(\displaystyle \boldsymbol{\phi}_{43} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 18 y \left(- 2 y z + y + 2 z - 1\right)\\\displaystyle 18 z \left(- 2 y z + 2 y + z - 1\right)\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{44}:\mathbf{v}\mapsto\displaystyle\int_{R}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 s_{1}\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(R\) is the reference element;
and \((s_{0},s_{1},s_{2})\) is a parametrisation of \(R(3)\).

\(\displaystyle \boldsymbol{\phi}_{44} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 30 y \left(- 2 y^{2} + 3 y - 1\right)\\\displaystyle 0\end{array}\right)\)

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

References

DefElement stats

Element added07 October 2021
Element last updated07 October 2021