an encyclopedia of finite element definitions

Tiniest tensor H(curl)

Click here to read what the information on this page means.

Alternative namesTNT H(curl)
Cockburn–fu names\(\left[S_{3,k}^\square\right]_{1}\)
Orders\(1\leqslant k\)
Reference elementsquadrilateral, hexahedron
Polynomial set\(\mathcal{Q}_{k}^d \oplus \mathcal{Z}^{(29)}_{k}\) (quadrilateral)
\(\mathcal{Q}_{k}^d \oplus \mathcal{Z}^{(30)}_{k} \oplus \mathcal{Z}^{(31)}_{k} \oplus \mathcal{Z}^{(32)}_{k}\) (hexahedron)
↓ Show polynomial set definitions ↓
DOFsOn each edge: tangent integral moments with an order \(k\) Lagrange space
On each face: integral moments with \(\nabla f\times \boldsymbol{n}\) (where \(\boldsymbol{n}\) is a unit vector normal to the face) for each \(f\) in an order \(k\) Lagrange space, and integral moments with \(\nabla\times\boldsymbol{f}\times \boldsymbol{n}\) (where \(\boldsymbol{n}\) is a unit vector normal to the face) for each \(\boldsymbol{f}\) in an order \(k\) vector Lagrange space such that \(\nabla\cdot\boldsymbol{f}=0\) and the normal trace of \(\boldsymbol{f}\) on the edges of the face is 0
On each volume: integral moments with \(\nabla\times\boldsymbol{f}\) for each \(\boldsymbol{f}\) in an order \(k\) vector Lagrange space such that the tangential trace of \(\boldsymbol{f}\) on the faces of the volume is 0, and integral moments with \(\nabla f\) for each \(f\) in an order \(k\) Lagrange space such that the trace of \(f\) on the faces of the volume is 0
Number of DOFsquadrilateral: \(2(k+1)^2 + 3\)
hexahedron: \(3(k+1)^3 + 18\)
CategoriesVector-valued elements, H(curl) conforming elements

Implementations

Symfem"TNTcurl" (quadrilateral, hexahedron)
↓ Show Symfem examples ↓

Examples

quadrilateral
order 1
quadrilateral
order 2
quadrilateral
order 3
hexahedron
order 1
  • \(R\) is the reference quadrilateral. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle \frac{3 y \left(1 - y\right)}{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{3 x \left(x - 1\right)}{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle \frac{3 y \left(2 x y - 2 x - y + 1\right)}{2}\\\displaystyle \frac{3 x \left(- 2 x y + x + 2 y - 1\right)}{2}\end{array}\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{10}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{0}\)
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{t}}_{0}\) is the tangent to edge 0;
and \(s_{0}\) is a parametrisation of \(e_{0}\).

\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle - 9 x y^{2} + 15 x y - 6 x + \frac{15 y^{2}}{2} - \frac{23 y}{2} + 4\\\displaystyle \frac{9 x \left(2 x y - x - 2 y + 1\right)}{2}\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 9 x y^{2} - 15 x y + 6 x - \frac{3 y^{2}}{2} + \frac{7 y}{2} - 2\\\displaystyle \frac{9 x \left(- 2 x y + x + 2 y - 1\right)}{2}\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle \frac{9 y \left(2 x y - 2 x - y + 1\right)}{2}\\\displaystyle - 9 x^{2} y + \frac{15 x^{2}}{2} + 15 x y - \frac{23 x}{2} - 6 y + 4\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle \frac{9 y \left(- 2 x y + 2 x + y - 1\right)}{2}\\\displaystyle 9 x^{2} y - \frac{3 x^{2}}{2} - 15 x y + \frac{7 x}{2} + 6 y - 2\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle \frac{9 y \left(2 x y - 2 x - y + 1\right)}{2}\\\displaystyle \frac{x \left(- 18 x y + 15 x + 6 y - 7\right)}{2}\end{array}\right)\)

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

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

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

\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle \frac{y \left(- 18 x y + 6 x + 15 y - 7\right)}{2}\\\displaystyle \frac{9 x \left(2 x y - x - 2 y + 1\right)}{2}\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle \frac{y \left(18 x y - 6 x - 3 y - 1\right)}{2}\\\displaystyle \frac{9 x \left(- 2 x y + x + 2 y - 1\right)}{2}\end{array}\right)\)

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.

\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 3 y \left(6 x y - 6 x - 5 y + 5\right)\\\displaystyle 9 x \left(- 2 x y + x + 2 y - 1\right)\end{array}\right)\)

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.

\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 9 y \left(2 x y - 2 x - y + 1\right)\\\displaystyle 3 x \left(- 6 x y + 5 x + 6 y - 5\right)\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{10}:\mathbf{v}\mapsto\displaystyle\int_{R}\left(\begin{array}{c}\displaystyle s_{0}\\\displaystyle - s_{1}\end{array}\right)\cdot\mathbf{v}\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).

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

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: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2} y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2} y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2} y^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2} y^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 2 y \left(2 y^{2} - 3 y + 1\right)\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 x \left(- 2 x^{2} + 3 x - 1\right)\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 2 y \left(12 x^{2} y^{2} - 18 x^{2} y + 6 x^{2} - 12 x y^{2} + 18 x y - 6 x + 2 y^{2} - 3 y + 1\right)\\\displaystyle 2 x \left(- 12 x^{2} y^{2} + 12 x^{2} y - 2 x^{2} + 18 x y^{2} - 18 x y + 3 x - 6 y^{2} + 6 y - 1\right)\end{array}\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{20}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(2 s_{0}^{2} - 3 s_{0} + 1)\hat{\boldsymbol{t}}_{0}\)
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{t}}_{0}\) is the tangent to edge 0;
and \(s_{0}\) is a parametrisation of \(e_{0}\).

\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle - 150 x^{2} y^{3} + 315 x^{2} y^{2} - 195 x^{2} y + 30 x^{2} + 150 x y^{3} - \frac{663 x y^{2}}{2} + \frac{435 x y}{2} - 36 x - 35 y^{3} + \frac{315 y^{2}}{4} - \frac{211 y}{4} + 9\\\displaystyle \frac{x \left(600 x^{2} y^{2} - 600 x^{2} y + 100 x^{2} - 1260 x y^{2} + 1266 x y - 213 x + 660 y^{2} - 666 y + 113\right)}{4}\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle - 150 x^{2} y^{3} + 315 x^{2} y^{2} - 195 x^{2} y + 30 x^{2} + 150 x y^{3} - \frac{597 x y^{2}}{2} + \frac{345 x y}{2} - 24 x - 35 y^{3} + \frac{249 y^{2}}{4} - \frac{121 y}{4} + 3\\\displaystyle \frac{x \left(600 x^{2} y^{2} - 600 x^{2} y + 100 x^{2} - 540 x y^{2} + 534 x y - 87 x - 60 y^{2} + 66 y - 13\right)}{4}\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 75 x^{2} y^{3} - \frac{315 x^{2} y^{2}}{2} + \frac{195 x^{2} y}{2} - 15 x^{2} - 75 x y^{3} + \frac{315 x y^{2}}{2} - \frac{195 x y}{2} + 15 x + \frac{5 y^{3}}{2} - \frac{33 y^{2}}{4} + \frac{29 y}{4} - \frac{3}{2}\\\displaystyle \frac{25 x \left(- 12 x^{2} y^{2} + 12 x^{2} y - 2 x^{2} + 18 x y^{2} - 18 x y + 3 x - 6 y^{2} + 6 y - 1\right)}{4}\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle \frac{y \left(600 x^{2} y^{2} - 1260 x^{2} y + 660 x^{2} - 600 x y^{2} + 1266 x y - 666 x + 100 y^{2} - 213 y + 113\right)}{4}\\\displaystyle - 150 x^{3} y^{2} + 150 x^{3} y - 35 x^{3} + 315 x^{2} y^{2} - \frac{663 x^{2} y}{2} + \frac{315 x^{2}}{4} - 195 x y^{2} + \frac{435 x y}{2} - \frac{211 x}{4} + 30 y^{2} - 36 y + 9\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle \frac{y \left(600 x^{2} y^{2} - 540 x^{2} y - 60 x^{2} - 600 x y^{2} + 534 x y + 66 x + 100 y^{2} - 87 y - 13\right)}{4}\\\displaystyle - 150 x^{3} y^{2} + 150 x^{3} y - 35 x^{3} + 315 x^{2} y^{2} - \frac{597 x^{2} y}{2} + \frac{249 x^{2}}{4} - 195 x y^{2} + \frac{345 x y}{2} - \frac{121 x}{4} + 30 y^{2} - 24 y + 3\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(4 s_{0} \left(1 - s_{0}\right))\hat{\boldsymbol{t}}_{1}\)
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{t}}_{1}\) is the tangent to edge 1;
and \(s_{0}\) is a parametrisation of \(e_{1}\).

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle \frac{25 y \left(- 12 x^{2} y^{2} + 18 x^{2} y - 6 x^{2} + 12 x y^{2} - 18 x y + 6 x - 2 y^{2} + 3 y - 1\right)}{4}\\\displaystyle 75 x^{3} y^{2} - 75 x^{3} y + \frac{5 x^{3}}{2} - \frac{315 x^{2} y^{2}}{2} + \frac{315 x^{2} y}{2} - \frac{33 x^{2}}{4} + \frac{195 x y^{2}}{2} - \frac{195 x y}{2} + \frac{29 x}{4} - 15 y^{2} + 15 y - \frac{3}{2}\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle \frac{y \left(- 600 x^{2} y^{2} + 1260 x^{2} y - 660 x^{2} + 600 x y^{2} - 1254 x y + 654 x - 100 y^{2} + 207 y - 107\right)}{4}\\\displaystyle \frac{x \left(600 x^{2} y^{2} - 600 x^{2} y + 140 x^{2} - 540 x y^{2} + 474 x y - 105 x + 60 y^{2} - 18 y + 1\right)}{4}\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle \frac{y \left(- 600 x^{2} y^{2} + 540 x^{2} y + 60 x^{2} + 600 x y^{2} - 546 x y - 54 x - 100 y^{2} + 93 y + 7\right)}{4}\\\displaystyle \frac{x \left(600 x^{2} y^{2} - 600 x^{2} y + 140 x^{2} - 540 x y^{2} + 606 x y - 171 x + 60 y^{2} - 102 y + 43\right)}{4}\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(4 s_{0} \left(1 - s_{0}\right))\hat{\boldsymbol{t}}_{2}\)
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{t}}_{2}\) is the tangent to edge 2;
and \(s_{0}\) is a parametrisation of \(e_{2}\).

\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle \frac{25 y \left(12 x^{2} y^{2} - 18 x^{2} y + 6 x^{2} - 12 x y^{2} + 18 x y - 6 x + 2 y^{2} - 3 y + 1\right)}{4}\\\displaystyle \frac{x \left(- 300 x^{2} y^{2} + 300 x^{2} y - 10 x^{2} + 270 x y^{2} - 270 x y - 3 x - 30 y^{2} + 30 y + 7\right)}{4}\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle \frac{y \left(600 x^{2} y^{2} - 540 x^{2} y + 60 x^{2} - 600 x y^{2} + 474 x y - 18 x + 140 y^{2} - 105 y + 1\right)}{4}\\\displaystyle \frac{x \left(- 600 x^{2} y^{2} + 600 x^{2} y - 100 x^{2} + 1260 x y^{2} - 1254 x y + 207 x - 660 y^{2} + 654 y - 107\right)}{4}\end{array}\right)\)

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{10}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(s_{0} \left(2 s_{0} - 1\right))\hat{\boldsymbol{t}}_{3}\)
where \(e_{3}\) is the 3th edge;
\(\hat{\boldsymbol{t}}_{3}\) is the tangent to edge 3;
and \(s_{0}\) is a parametrisation of \(e_{3}\).

\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle \frac{y \left(600 x^{2} y^{2} - 540 x^{2} y + 60 x^{2} - 600 x y^{2} + 606 x y - 102 x + 140 y^{2} - 171 y + 43\right)}{4}\\\displaystyle \frac{x \left(- 600 x^{2} y^{2} + 600 x^{2} y - 100 x^{2} + 540 x y^{2} - 546 x y + 93 x + 60 y^{2} - 54 y + 7\right)}{4}\end{array}\right)\)

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{11}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(4 s_{0} \left(1 - s_{0}\right))\hat{\boldsymbol{t}}_{3}\)
where \(e_{3}\) is the 3th edge;
\(\hat{\boldsymbol{t}}_{3}\) is the tangent to edge 3;
and \(s_{0}\) is a parametrisation of \(e_{3}\).

\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle \frac{y \left(- 300 x^{2} y^{2} + 270 x^{2} y - 30 x^{2} + 300 x y^{2} - 270 x y + 30 x - 10 y^{2} - 3 y + 7\right)}{4}\\\displaystyle \frac{25 x \left(12 x^{2} y^{2} - 12 x^{2} y + 2 x^{2} - 18 x y^{2} + 18 x y - 3 x + 6 y^{2} - 6 y + 1\right)}{4}\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 6 y \left(150 x^{2} y^{2} - 255 x^{2} y + 105 x^{2} - 150 x y^{2} + 258 x y - 108 x + 35 y^{2} - 60 y + 25\right)\\\displaystyle 6 x \left(- 150 x^{2} y^{2} + 150 x^{2} y - 25 x^{2} + 315 x y^{2} - 318 x y + 54 x - 165 y^{2} + 168 y - 29\right)\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{13}:\mathbf{v}\mapsto\displaystyle\int_{R}\left(\begin{array}{c}\displaystyle 2 s_{1}\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).

\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 15 y \left(- 60 x^{2} y^{2} + 90 x^{2} y - 30 x^{2} + 60 x y^{2} - 90 x y + 30 x - 14 y^{2} + 21 y - 7\right)\\\displaystyle 15 x \left(60 x^{2} y^{2} - 60 x^{2} y + 10 x^{2} - 126 x y^{2} + 126 x y - 21 x + 66 y^{2} - 66 y + 11\right)\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 6 y \left(150 x^{2} y^{2} - 315 x^{2} y + 165 x^{2} - 150 x y^{2} + 318 x y - 168 x + 25 y^{2} - 54 y + 29\right)\\\displaystyle 6 x \left(- 150 x^{2} y^{2} + 150 x^{2} y - 35 x^{2} + 255 x y^{2} - 258 x y + 60 x - 105 y^{2} + 108 y - 25\right)\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 36 y \left(- 150 x^{2} y^{2} + 255 x^{2} y - 105 x^{2} + 150 x y^{2} - 256 x y + 106 x - 25 y^{2} + 43 y - 18\right)\\\displaystyle 36 x \left(150 x^{2} y^{2} - 150 x^{2} y + 25 x^{2} - 255 x y^{2} + 256 x y - 43 x + 105 y^{2} - 106 y + 18\right)\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle 450 y \left(12 x^{2} y^{2} - 18 x^{2} y + 6 x^{2} - 12 x y^{2} + 18 x y - 6 x + 2 y^{2} - 3 y + 1\right)\\\displaystyle 90 x \left(- 60 x^{2} y^{2} + 60 x^{2} y - 10 x^{2} + 102 x y^{2} - 102 x y + 17 x - 42 y^{2} + 42 y - 7\right)\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{17}:\mathbf{v}\mapsto\displaystyle\int_{R}\left(\begin{array}{c}\displaystyle 0\\\displaystyle - 2 s_{0}\end{array}\right)\cdot\mathbf{v}\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).

\(\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle 15 y \left(- 60 x^{2} y^{2} + 126 x^{2} y - 66 x^{2} + 60 x y^{2} - 126 x y + 66 x - 10 y^{2} + 21 y - 11\right)\\\displaystyle 15 x \left(60 x^{2} y^{2} - 60 x^{2} y + 14 x^{2} - 90 x y^{2} + 90 x y - 21 x + 30 y^{2} - 30 y + 7\right)\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{18}:\mathbf{v}\mapsto\displaystyle\int_{R}\left(\begin{array}{c}\displaystyle s_{0}^{2}\\\displaystyle - 2 s_{0} s_{1}\end{array}\right)\cdot\mathbf{v}\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).

\(\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle 90 y \left(60 x^{2} y^{2} - 102 x^{2} y + 42 x^{2} - 60 x y^{2} + 102 x y - 42 x + 10 y^{2} - 17 y + 7\right)\\\displaystyle 450 x \left(- 12 x^{2} y^{2} + 12 x^{2} y - 2 x^{2} + 18 x y^{2} - 18 x y + 3 x - 6 y^{2} + 6 y - 1\right)\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{19}:\mathbf{v}\mapsto\displaystyle\int_{R}\left(\begin{array}{c}\displaystyle 2 s_{0}^{2} s_{1}\\\displaystyle - 2 s_{0} s_{1}^{2}\end{array}\right)\cdot\mathbf{v}\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).

\(\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle 450 y \left(- 12 x^{2} y^{2} + 18 x^{2} y - 6 x^{2} + 12 x y^{2} - 18 x y + 6 x - 2 y^{2} + 3 y - 1\right)\\\displaystyle 450 x \left(12 x^{2} y^{2} - 12 x^{2} y + 2 x^{2} - 18 x y^{2} + 18 x y - 3 x + 6 y^{2} - 6 y + 1\right)\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{20}:\mathbf{v}\mapsto\displaystyle\int_{R}\left(\begin{array}{c}\displaystyle s_{1} \left(- 2 s_{0} s_{1} + 2 s_{0} + s_{1} - 1\right)\\\displaystyle s_{0} \left(- 2 s_{0} s_{1} + s_{0} + 2 s_{1} - 1\right)\end{array}\right)\cdot\mathbf{v}\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).

\(\displaystyle \boldsymbol{\phi}_{20} = \left(\begin{array}{c}\displaystyle 45 y \left(- 2 x y + 2 x + y - 1\right)\\\displaystyle 45 x \left(- 2 x y + x + 2 y - 1\right)\end{array}\right)\)

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: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y^{3}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{3}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y^{3}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y^{3}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2} y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2} y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2} y^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2} y^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2} y^{3}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2} y^{3}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{3}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{3}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{3} y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{3} y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{3} y^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{3} y^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{3} y^{3}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{3} y^{3}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle \frac{7 y \left(- 5 y^{3} + 10 y^{2} - 6 y + 1\right)}{3}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{7 x \left(5 x^{3} - 10 x^{2} + 6 x - 1\right)}{3}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle \frac{7 y \left(100 x^{3} y^{3} - 200 x^{3} y^{2} + 120 x^{3} y - 20 x^{3} - 150 x^{2} y^{3} + 300 x^{2} y^{2} - 180 x^{2} y + 30 x^{2} + 60 x y^{3} - 120 x y^{2} + 72 x y - 12 x - 5 y^{3} + 10 y^{2} - 6 y + 1\right)}{3}\\\displaystyle \frac{7 x \left(- 100 x^{3} y^{3} + 150 x^{3} y^{2} - 60 x^{3} y + 5 x^{3} + 200 x^{2} y^{3} - 300 x^{2} y^{2} + 120 x^{2} y - 10 x^{2} - 120 x y^{3} + 180 x y^{2} - 72 x y + 6 x + 20 y^{3} - 30 y^{2} + 12 y - 1\right)}{3}\end{array}\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{34}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(- \frac{9 s_{0}^{3}}{2} + 9 s_{0}^{2} - \frac{11 s_{0}}{2} + 1)\hat{\boldsymbol{t}}_{0}\)
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{t}}_{0}\) is the tangent to edge 0;
and \(s_{0}\) is a parametrisation of \(e_{0}\).

\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle - 2450 x^{3} y^{4} + 6300 x^{3} y^{3} - 5460 x^{3} y^{2} + 1750 x^{3} y - 140 x^{3} + 3675 x^{2} y^{4} - 9705 x^{2} y^{3} + \frac{112455 x^{2} y^{2}}{13} - \frac{37185 x^{2} y}{13} + 240 x^{2} - 1470 x y^{4} + \frac{53190 x y^{3}}{13} - \frac{49890 x y^{2}}{13} + \frac{17370 x y}{13} - 120 x + \frac{315 y^{4}}{2} - \frac{6055 y^{3}}{13} + \frac{5985 y^{2}}{13} - \frac{4371 y}{26} + 16\\\displaystyle \frac{5 x \left(12740 x^{3} y^{3} - 19110 x^{3} y^{2} + 7644 x^{3} y - 637 x^{3} - 32760 x^{2} y^{3} + 49374 x^{2} y^{2} - 19932 x^{2} y + 1698 x^{2} + 28392 x y^{3} - 42966 x y^{2} + 17484 x y - 1518 x - 8372 y^{3} + 12702 y^{2} - 5196 y + 457\right)}{26}\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 2450 x^{3} y^{4} - 6300 x^{3} y^{3} + 5460 x^{3} y^{2} - 1750 x^{3} y + 140 x^{3} - 3675 x^{2} y^{4} + 9195 x^{2} y^{3} - \frac{100485 x^{2} y^{2}}{13} + \frac{31065 x^{2} y}{13} - 180 x^{2} + 1470 x y^{4} - \frac{46560 x y^{3}}{13} + \frac{37920 x y^{2}}{13} - \frac{11250 x y}{13} + 60 x - \frac{175 y^{4}}{2} + \frac{2870 y^{3}}{13} - \frac{2430 y^{2}}{13} + \frac{1499 y}{26} - 4\\\displaystyle \frac{5 x \left(- 12740 x^{3} y^{3} + 19110 x^{3} y^{2} - 7644 x^{3} y + 637 x^{3} + 18200 x^{2} y^{3} - 27066 x^{2} y^{2} + 10644 x^{2} y - 850 x^{2} - 6552 x y^{3} + 9504 x y^{2} - 3552 x y + 246 x + 1092 y^{3} - 1548 y^{2} + 552 y - 33\right)}{26}\end{array}\right)\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(\frac{9 s_{0} \left(3 s_{0}^{2} - 5 s_{0} + 2\right)}{2})\hat{\boldsymbol{t}}_{0}\)
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{t}}_{0}\) is the tangent to edge 0;
and \(s_{0}\) is a parametrisation of \(e_{0}\).

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle \frac{26950 x^{3} y^{4}}{27} - \frac{7700 x^{3} y^{3}}{3} + \frac{20020 x^{3} y^{2}}{9} - \frac{19250 x^{3} y}{27} + \frac{1540 x^{3}}{27} - \frac{13475 x^{2} y^{4}}{9} + 3935 x^{2} y^{3} - \frac{136115 x^{2} y^{2}}{39} + \frac{134305 x^{2} y}{117} - \frac{860 x^{2}}{9} + \frac{5390 x y^{4}}{9} - \frac{20880 x y^{3}}{13} + \frac{56720 x y^{2}}{39} - \frac{57250 x y}{117} + \frac{380 x}{9} - \frac{805 y^{4}}{54} + \frac{690 y^{3}}{13} - \frac{7010 y^{2}}{117} + \frac{17033 y}{702} - \frac{68}{27}\\\displaystyle \frac{5 x \left(- 140140 x^{3} y^{3} + 210210 x^{3} y^{2} - 84084 x^{3} y + 7007 x^{3} + 345800 x^{2} y^{3} - 520806 x^{2} y^{2} + 209964 x^{2} y - 17830 x^{2} - 246792 x y^{3} + 373104 x y^{2} - 151488 x y + 13074 x + 41132 y^{3} - 62508 y^{2} + 25608 y - 2251\right)}{702}\end{array}\right)\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(\frac{9 s_{0} \left(- 3 s_{0}^{2} + 4 s_{0} - 1\right)}{2})\hat{\boldsymbol{t}}_{0}\)
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{t}}_{0}\) is the tangent to edge 0;
and \(s_{0}\) is a parametrisation of \(e_{0}\).

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle - \frac{26950 x^{3} y^{4}}{27} + \frac{7700 x^{3} y^{3}}{3} - \frac{20020 x^{3} y^{2}}{9} + \frac{19250 x^{3} y}{27} - \frac{1540 x^{3}}{27} + \frac{13475 x^{2} y^{4}}{9} - 3765 x^{2} y^{3} + \frac{124145 x^{2} y^{2}}{39} - \frac{115945 x^{2} y}{117} + \frac{680 x^{2}}{9} - \frac{5390 x y^{4}}{9} + \frac{18670 x y^{3}}{13} - \frac{44750 x y^{2}}{39} + \frac{38890 x y}{117} - \frac{200 x}{9} + \frac{4585 y^{4}}{54} - \frac{7205 y^{3}}{39} + \frac{15065 y^{2}}{117} - \frac{21137 y}{702} + \frac{32}{27}\\\displaystyle \frac{5 x \left(140140 x^{3} y^{3} - 210210 x^{3} y^{2} + 84084 x^{3} y - 7007 x^{3} - 214760 x^{2} y^{3} + 320034 x^{2} y^{2} - 126372 x^{2} y + 10198 x^{2} + 50232 x y^{3} - 71946 x y^{2} + 26100 x y - 1626 x + 24388 y^{3} - 37878 y^{2} + 16188 y - 1565\right)}{702}\end{array}\right)\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(- \frac{9 s_{0}^{3}}{2} + 9 s_{0}^{2} - \frac{11 s_{0}}{2} + 1)\hat{\boldsymbol{t}}_{1}\)
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{t}}_{1}\) is the tangent to edge 1;
and \(s_{0}\) is a parametrisation of \(e_{1}\).

\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle \frac{5 y \left(12740 x^{3} y^{3} - 32760 x^{3} y^{2} + 28392 x^{3} y - 8372 x^{3} - 19110 x^{2} y^{3} + 49374 x^{2} y^{2} - 42966 x^{2} y + 12702 x^{2} + 7644 x y^{3} - 19932 x y^{2} + 17484 x y - 5196 x - 637 y^{3} + 1698 y^{2} - 1518 y + 457\right)}{26}\\\displaystyle - 2450 x^{4} y^{3} + 3675 x^{4} y^{2} - 1470 x^{4} y + \frac{315 x^{4}}{2} + 6300 x^{3} y^{3} - 9705 x^{3} y^{2} + \frac{53190 x^{3} y}{13} - \frac{6055 x^{3}}{13} - 5460 x^{2} y^{3} + \frac{112455 x^{2} y^{2}}{13} - \frac{49890 x^{2} y}{13} + \frac{5985 x^{2}}{13} + 1750 x y^{3} - \frac{37185 x y^{2}}{13} + \frac{17370 x y}{13} - \frac{4371 x}{26} - 140 y^{3} + 240 y^{2} - 120 y + 16\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(\frac{s_{0} \left(9 s_{0}^{2} - 9 s_{0} + 2\right)}{2})\hat{\boldsymbol{t}}_{1}\)
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{t}}_{1}\) is the tangent to edge 1;
and \(s_{0}\) is a parametrisation of \(e_{1}\).

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle \frac{5 y \left(- 12740 x^{3} y^{3} + 18200 x^{3} y^{2} - 6552 x^{3} y + 1092 x^{3} + 19110 x^{2} y^{3} - 27066 x^{2} y^{2} + 9504 x^{2} y - 1548 x^{2} - 7644 x y^{3} + 10644 x y^{2} - 3552 x y + 552 x + 637 y^{3} - 850 y^{2} + 246 y - 33\right)}{26}\\\displaystyle 2450 x^{4} y^{3} - 3675 x^{4} y^{2} + 1470 x^{4} y - \frac{175 x^{4}}{2} - 6300 x^{3} y^{3} + 9195 x^{3} y^{2} - \frac{46560 x^{3} y}{13} + \frac{2870 x^{3}}{13} + 5460 x^{2} y^{3} - \frac{100485 x^{2} y^{2}}{13} + \frac{37920 x^{2} y}{13} - \frac{2430 x^{2}}{13} - 1750 x y^{3} + \frac{31065 x y^{2}}{13} - \frac{11250 x y}{13} + \frac{1499 x}{26} + 140 y^{3} - 180 y^{2} + 60 y - 4\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(\frac{9 s_{0} \left(3 s_{0}^{2} - 5 s_{0} + 2\right)}{2})\hat{\boldsymbol{t}}_{1}\)
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{t}}_{1}\) is the tangent to edge 1;
and \(s_{0}\) is a parametrisation of \(e_{1}\).

\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle \frac{5 y \left(- 140140 x^{3} y^{3} + 345800 x^{3} y^{2} - 246792 x^{3} y + 41132 x^{3} + 210210 x^{2} y^{3} - 520806 x^{2} y^{2} + 373104 x^{2} y - 62508 x^{2} - 84084 x y^{3} + 209964 x y^{2} - 151488 x y + 25608 x + 7007 y^{3} - 17830 y^{2} + 13074 y - 2251\right)}{702}\\\displaystyle \frac{26950 x^{4} y^{3}}{27} - \frac{13475 x^{4} y^{2}}{9} + \frac{5390 x^{4} y}{9} - \frac{805 x^{4}}{54} - \frac{7700 x^{3} y^{3}}{3} + 3935 x^{3} y^{2} - \frac{20880 x^{3} y}{13} + \frac{690 x^{3}}{13} + \frac{20020 x^{2} y^{3}}{9} - \frac{136115 x^{2} y^{2}}{39} + \frac{56720 x^{2} y}{39} - \frac{7010 x^{2}}{117} - \frac{19250 x y^{3}}{27} + \frac{134305 x y^{2}}{117} - \frac{57250 x y}{117} + \frac{17033 x}{702} + \frac{1540 y^{3}}{27} - \frac{860 y^{2}}{9} + \frac{380 y}{9} - \frac{68}{27}\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(\frac{9 s_{0} \left(- 3 s_{0}^{2} + 4 s_{0} - 1\right)}{2})\hat{\boldsymbol{t}}_{1}\)
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{t}}_{1}\) is the tangent to edge 1;
and \(s_{0}\) is a parametrisation of \(e_{1}\).

\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle \frac{5 y \left(140140 x^{3} y^{3} - 214760 x^{3} y^{2} + 50232 x^{3} y + 24388 x^{3} - 210210 x^{2} y^{3} + 320034 x^{2} y^{2} - 71946 x^{2} y - 37878 x^{2} + 84084 x y^{3} - 126372 x y^{2} + 26100 x y + 16188 x - 7007 y^{3} + 10198 y^{2} - 1626 y - 1565\right)}{702}\\\displaystyle - \frac{26950 x^{4} y^{3}}{27} + \frac{13475 x^{4} y^{2}}{9} - \frac{5390 x^{4} y}{9} + \frac{4585 x^{4}}{54} + \frac{7700 x^{3} y^{3}}{3} - 3765 x^{3} y^{2} + \frac{18670 x^{3} y}{13} - \frac{7205 x^{3}}{39} - \frac{20020 x^{2} y^{3}}{9} + \frac{124145 x^{2} y^{2}}{39} - \frac{44750 x^{2} y}{39} + \frac{15065 x^{2}}{117} + \frac{19250 x y^{3}}{27} - \frac{115945 x y^{2}}{117} + \frac{38890 x y}{117} - \frac{21137 x}{702} - \frac{1540 y^{3}}{27} + \frac{680 y^{2}}{9} - \frac{200 y}{9} + \frac{32}{27}\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(- \frac{9 s_{0}^{3}}{2} + 9 s_{0}^{2} - \frac{11 s_{0}}{2} + 1)\hat{\boldsymbol{t}}_{2}\)
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{t}}_{2}\) is the tangent to edge 2;
and \(s_{0}\) is a parametrisation of \(e_{2}\).

\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle \frac{5 y \left(12740 x^{3} y^{3} - 32760 x^{3} y^{2} + 28392 x^{3} y - 8372 x^{3} - 19110 x^{2} y^{3} + 48906 x^{2} y^{2} - 42210 x^{2} y + 12414 x^{2} + 7644 x y^{3} - 19464 x y^{2} + 16728 x y - 4908 x - 637 y^{3} + 1620 y^{2} - 1392 y + 409\right)}{26}\\\displaystyle \frac{x \left(- 63700 x^{3} y^{3} + 95550 x^{3} y^{2} - 38220 x^{3} y + 4095 x^{3} + 91000 x^{2} y^{3} - 129870 x^{2} y^{2} + 46500 x^{2} y - 4270 x^{2} - 32760 x y^{3} + 41220 x y^{2} - 9960 x y + 210 x + 1820 y^{3} - 660 y^{2} - 1440 y + 381\right)}{26}\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{9}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(\frac{s_{0} \left(9 s_{0}^{2} - 9 s_{0} + 2\right)}{2})\hat{\boldsymbol{t}}_{2}\)
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{t}}_{2}\) is the tangent to edge 2;
and \(s_{0}\) is a parametrisation of \(e_{2}\).

\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle \frac{5 y \left(- 12740 x^{3} y^{3} + 18200 x^{3} y^{2} - 6552 x^{3} y + 1092 x^{3} + 19110 x^{2} y^{3} - 27534 x^{2} y^{2} + 10152 x^{2} y - 1728 x^{2} - 7644 x y^{3} + 11112 x y^{2} - 4200 x y + 732 x + 637 y^{3} - 928 y^{2} + 354 y - 63\right)}{26}\\\displaystyle \frac{x \left(63700 x^{3} y^{3} - 95550 x^{3} y^{2} + 38220 x^{3} y - 2275 x^{3} - 91000 x^{2} y^{3} + 143130 x^{2} y^{2} - 59760 x^{2} y + 3360 x^{2} + 32760 x y^{3} - 57060 x y^{2} + 25800 x y - 1290 x - 1820 y^{3} + 4800 y^{2} - 2700 y + 101\right)}{26}\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{10}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(\frac{9 s_{0} \left(3 s_{0}^{2} - 5 s_{0} + 2\right)}{2})\hat{\boldsymbol{t}}_{2}\)
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{t}}_{2}\) is the tangent to edge 2;
and \(s_{0}\) is a parametrisation of \(e_{2}\).

\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle \frac{5 y \left(- 140140 x^{3} y^{3} + 345800 x^{3} y^{2} - 246792 x^{3} y + 41132 x^{3} + 210210 x^{2} y^{3} - 516594 x^{2} y^{2} + 367272 x^{2} y - 60888 x^{2} - 84084 x y^{3} + 205752 x y^{2} - 145656 x y + 23988 x + 7007 y^{3} - 17128 y^{2} + 12102 y - 1981\right)}{702}\\\displaystyle \frac{x \left(700700 x^{3} y^{3} - 1051050 x^{3} y^{2} + 420420 x^{3} y - 10465 x^{3} - 1001000 x^{2} y^{3} + 1441830 x^{2} y^{2} - 554160 x^{2} y + 4600 x^{2} + 360360 x y^{3} - 469260 x y^{2} + 160920 x y + 6930 x - 20020 y^{3} + 11400 y^{2} + 2460 y - 2833\right)}{702}\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{11}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(\frac{9 s_{0} \left(- 3 s_{0}^{2} + 4 s_{0} - 1\right)}{2})\hat{\boldsymbol{t}}_{2}\)
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{t}}_{2}\) is the tangent to edge 2;
and \(s_{0}\) is a parametrisation of \(e_{2}\).

\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle \frac{5 y \left(140140 x^{3} y^{3} - 214760 x^{3} y^{2} + 50232 x^{3} y + 24388 x^{3} - 210210 x^{2} y^{3} + 324246 x^{2} y^{2} - 78750 x^{2} y - 35286 x^{2} + 84084 x y^{3} - 130584 x y^{2} + 32904 x y + 13596 x - 7007 y^{3} + 10900 y^{2} - 2760 y - 1133\right)}{702}\\\displaystyle \frac{x \left(- 700700 x^{3} y^{3} + 1051050 x^{3} y^{2} - 420420 x^{3} y + 59605 x^{3} + 1001000 x^{2} y^{3} - 1561170 x^{2} y^{2} + 673500 x^{2} y - 108730 x^{2} - 360360 x y^{3} + 611820 x y^{2} - 303480 x y + 58950 x + 20020 y^{3} - 48660 y^{2} + 34800 y - 8993\right)}{702}\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{12}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(- \frac{9 s_{0}^{3}}{2} + 9 s_{0}^{2} - \frac{11 s_{0}}{2} + 1)\hat{\boldsymbol{t}}_{3}\)
where \(e_{3}\) is the 3th edge;
\(\hat{\boldsymbol{t}}_{3}\) is the tangent to edge 3;
and \(s_{0}\) is a parametrisation of \(e_{3}\).

\(\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle \frac{y \left(- 63700 x^{3} y^{3} + 91000 x^{3} y^{2} - 32760 x^{3} y + 1820 x^{3} + 95550 x^{2} y^{3} - 129870 x^{2} y^{2} + 41220 x^{2} y - 660 x^{2} - 38220 x y^{3} + 46500 x y^{2} - 9960 x y - 1440 x + 4095 y^{3} - 4270 y^{2} + 210 y + 381\right)}{26}\\\displaystyle \frac{5 x \left(12740 x^{3} y^{3} - 19110 x^{3} y^{2} + 7644 x^{3} y - 637 x^{3} - 32760 x^{2} y^{3} + 48906 x^{2} y^{2} - 19464 x^{2} y + 1620 x^{2} + 28392 x y^{3} - 42210 x y^{2} + 16728 x y - 1392 x - 8372 y^{3} + 12414 y^{2} - 4908 y + 409\right)}{26}\end{array}\right)\)

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(\frac{s_{0} \left(9 s_{0}^{2} - 9 s_{0} + 2\right)}{2})\hat{\boldsymbol{t}}_{3}\)
where \(e_{3}\) is the 3th edge;
\(\hat{\boldsymbol{t}}_{3}\) is the tangent to edge 3;
and \(s_{0}\) is a parametrisation of \(e_{3}\).

\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle \frac{y \left(63700 x^{3} y^{3} - 91000 x^{3} y^{2} + 32760 x^{3} y - 1820 x^{3} - 95550 x^{2} y^{3} + 143130 x^{2} y^{2} - 57060 x^{2} y + 4800 x^{2} + 38220 x y^{3} - 59760 x y^{2} + 25800 x y - 2700 x - 2275 y^{3} + 3360 y^{2} - 1290 y + 101\right)}{26}\\\displaystyle \frac{5 x \left(- 12740 x^{3} y^{3} + 19110 x^{3} y^{2} - 7644 x^{3} y + 637 x^{3} + 18200 x^{2} y^{3} - 27534 x^{2} y^{2} + 11112 x^{2} y - 928 x^{2} - 6552 x y^{3} + 10152 x y^{2} - 4200 x y + 354 x + 1092 y^{3} - 1728 y^{2} + 732 y - 63\right)}{26}\end{array}\right)\)

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{14}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(\frac{9 s_{0} \left(3 s_{0}^{2} - 5 s_{0} + 2\right)}{2})\hat{\boldsymbol{t}}_{3}\)
where \(e_{3}\) is the 3th edge;
\(\hat{\boldsymbol{t}}_{3}\) is the tangent to edge 3;
and \(s_{0}\) is a parametrisation of \(e_{3}\).

\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle \frac{y \left(700700 x^{3} y^{3} - 1001000 x^{3} y^{2} + 360360 x^{3} y - 20020 x^{3} - 1051050 x^{2} y^{3} + 1441830 x^{2} y^{2} - 469260 x^{2} y + 11400 x^{2} + 420420 x y^{3} - 554160 x y^{2} + 160920 x y + 2460 x - 10465 y^{3} + 4600 y^{2} + 6930 y - 2833\right)}{702}\\\displaystyle \frac{5 x \left(- 140140 x^{3} y^{3} + 210210 x^{3} y^{2} - 84084 x^{3} y + 7007 x^{3} + 345800 x^{2} y^{3} - 516594 x^{2} y^{2} + 205752 x^{2} y - 17128 x^{2} - 246792 x y^{3} + 367272 x y^{2} - 145656 x y + 12102 x + 41132 y^{3} - 60888 y^{2} + 23988 y - 1981\right)}{702}\end{array}\right)\)

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{15}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(\frac{9 s_{0} \left(- 3 s_{0}^{2} + 4 s_{0} - 1\right)}{2})\hat{\boldsymbol{t}}_{3}\)
where \(e_{3}\) is the 3th edge;
\(\hat{\boldsymbol{t}}_{3}\) is the tangent to edge 3;
and \(s_{0}\) is a parametrisation of \(e_{3}\).

\(\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle \frac{y \left(- 700700 x^{3} y^{3} + 1001000 x^{3} y^{2} - 360360 x^{3} y + 20020 x^{3} + 1051050 x^{2} y^{3} - 1561170 x^{2} y^{2} + 611820 x^{2} y - 48660 x^{2} - 420420 x y^{3} + 673500 x y^{2} - 303480 x y + 34800 x + 59605 y^{3} - 108730 y^{2} + 58950 y - 8993\right)}{702}\\\displaystyle \frac{5 x \left(140140 x^{3} y^{3} - 210210 x^{3} y^{2} + 84084 x^{3} y - 7007 x^{3} - 214760 x^{2} y^{3} + 324246 x^{2} y^{2} - 130584 x^{2} y + 10900 x^{2} + 50232 x y^{3} - 78750 x y^{2} + 32904 x y - 2760 x + 24388 y^{3} - 35286 y^{2} + 13596 y - 1133\right)}{702}\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle \frac{30 y \left(12740 x^{3} y^{3} - 29120 x^{3} y^{2} + 21112 x^{3} y - 4732 x^{3} - 19110 x^{2} y^{3} + 44070 x^{2} y^{2} - 32268 x^{2} y + 7308 x^{2} + 7644 x y^{3} - 17988 x y^{2} + 13464 x y - 3120 x - 819 y^{3} + 1974 y^{2} - 1514 y + 359\right)}{13}\\\displaystyle \frac{30 x \left(- 12740 x^{3} y^{3} + 19110 x^{3} y^{2} - 7644 x^{3} y + 637 x^{3} + 32760 x^{2} y^{3} - 49530 x^{2} y^{2} + 20088 x^{2} y - 1724 x^{2} - 28392 x y^{3} + 43218 x y^{2} - 17736 x y + 1560 x + 8372 y^{3} - 12798 y^{2} + 5292 y - 473\right)}{13}\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{17}:\mathbf{v}\mapsto\displaystyle\int_{R}\left(\begin{array}{c}\displaystyle 2 s_{1}\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).

\(\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle \frac{15 y \left(- 63700 x^{3} y^{3} + 134680 x^{3} y^{2} - 87360 x^{3} y + 16380 x^{3} + 95550 x^{2} y^{3} - 202800 x^{2} y^{2} + 132210 x^{2} y - 24960 x^{2} - 38220 x y^{3} + 81840 x y^{2} - 53964 x y + 10344 x + 4095 y^{3} - 8862 y^{2} + 5922 y - 1155\right)}{13}\\\displaystyle \frac{15 x \left(63700 x^{3} y^{3} - 95550 x^{3} y^{2} + 38220 x^{3} y - 3185 x^{3} - 163800 x^{2} y^{3} + 246480 x^{2} y^{2} - 99060 x^{2} y + 8320 x^{2} + 141960 x y^{3} - 214200 x y^{2} + 86436 x y - 7308 x - 41860 y^{3} + 63270 y^{2} - 25596 y + 2173\right)}{13}\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{18}:\mathbf{v}\mapsto\displaystyle\int_{R}\left(\begin{array}{c}\displaystyle 3 s_{1}^{2}\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).

\(\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle 70 y \left(700 x^{3} y^{3} - 1400 x^{3} y^{2} + 840 x^{3} y - 140 x^{3} - 1050 x^{2} y^{3} + 2100 x^{2} y^{2} - 1260 x^{2} y + 210 x^{2} + 420 x y^{3} - 840 x y^{2} + 504 x y - 84 x - 45 y^{3} + 90 y^{2} - 54 y + 9\right)\\\displaystyle 70 x \left(- 700 x^{3} y^{3} + 1050 x^{3} y^{2} - 420 x^{3} y + 35 x^{3} + 1800 x^{2} y^{3} - 2700 x^{2} y^{2} + 1080 x^{2} y - 90 x^{2} - 1560 x y^{3} + 2340 x y^{2} - 936 x y + 78 x + 460 y^{3} - 690 y^{2} + 276 y - 23\right)\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle \frac{30 y \left(12740 x^{3} y^{3} - 32760 x^{3} y^{2} + 28392 x^{3} y - 8372 x^{3} - 19110 x^{2} y^{3} + 49530 x^{2} y^{2} - 43218 x^{2} y + 12798 x^{2} + 7644 x y^{3} - 20088 x y^{2} + 17736 x y - 5292 x - 637 y^{3} + 1724 y^{2} - 1560 y + 473\right)}{13}\\\displaystyle \frac{30 x \left(- 12740 x^{3} y^{3} + 19110 x^{3} y^{2} - 7644 x^{3} y + 819 x^{3} + 29120 x^{2} y^{3} - 44070 x^{2} y^{2} + 17988 x^{2} y - 1974 x^{2} - 21112 x y^{3} + 32268 x y^{2} - 13464 x y + 1514 x + 4732 y^{3} - 7308 y^{2} + 3120 y - 359\right)}{13}\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{20}:\mathbf{v}\mapsto\displaystyle\int_{R}\left(\begin{array}{c}\displaystyle s_{0}\\\displaystyle - s_{1}\end{array}\right)\cdot\mathbf{v}\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).

\(\displaystyle \boldsymbol{\phi}_{20} = \left(\begin{array}{c}\displaystyle \frac{360 y \left(- 12740 x^{3} y^{3} + 29120 x^{3} y^{2} - 21112 x^{3} y + 4732 x^{3} + 19110 x^{2} y^{3} - 43875 x^{2} y^{2} + 31968 x^{2} y - 7203 x^{2} - 7644 x y^{3} + 17688 x y^{2} - 13000 x y + 2956 x + 637 y^{3} - 1499 y^{2} + 1122 y - 260\right)}{13}\\\displaystyle \frac{360 x \left(12740 x^{3} y^{3} - 19110 x^{3} y^{2} + 7644 x^{3} y - 637 x^{3} - 29120 x^{2} y^{3} + 43875 x^{2} y^{2} - 17688 x^{2} y + 1499 x^{2} + 21112 x y^{3} - 31968 x y^{2} + 13000 x y - 1122 x - 4732 y^{3} + 7203 y^{2} - 2956 y + 260\right)}{13}\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{21}:\mathbf{v}\mapsto\displaystyle\int_{R}\left(\begin{array}{c}\displaystyle 2 s_{0} s_{1}\\\displaystyle - s_{1}^{2}\end{array}\right)\cdot\mathbf{v}\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).

\(\displaystyle \boldsymbol{\phi}_{21} = \left(\begin{array}{c}\displaystyle \frac{180 y \left(63700 x^{3} y^{3} - 134680 x^{3} y^{2} + 87360 x^{3} y - 16380 x^{3} - 95550 x^{2} y^{3} + 202410 x^{2} y^{2} - 131625 x^{2} y + 24765 x^{2} + 38220 x y^{3} - 81240 x y^{2} + 53064 x y - 10044 x - 3185 y^{3} + 6820 y^{2} - 4497 y + 862\right)}{13}\\\displaystyle \frac{180 x \left(- 63700 x^{3} y^{3} + 95550 x^{3} y^{2} - 38220 x^{3} y + 3185 x^{3} + 145600 x^{2} y^{3} - 218790 x^{2} y^{2} + 87750 x^{2} y - 7345 x^{2} - 105560 x y^{3} + 158940 x y^{2} - 63936 x y + 5378 x + 23660 y^{3} - 35700 y^{2} + 14406 y - 1218\right)}{13}\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{22}:\mathbf{v}\mapsto\displaystyle\int_{R}\left(\begin{array}{c}\displaystyle 3 s_{0} s_{1}^{2}\\\displaystyle - s_{1}^{3}\end{array}\right)\cdot\mathbf{v}\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).

\(\displaystyle \boldsymbol{\phi}_{22} = \left(\begin{array}{c}\displaystyle 5880 y \left(- 100 x^{3} y^{3} + 200 x^{3} y^{2} - 120 x^{3} y + 20 x^{3} + 150 x^{2} y^{3} - 300 x^{2} y^{2} + 180 x^{2} y - 30 x^{2} - 60 x y^{3} + 120 x y^{2} - 72 x y + 12 x + 5 y^{3} - 10 y^{2} + 6 y - 1\right)\\\displaystyle 840 x \left(700 x^{3} y^{3} - 1050 x^{3} y^{2} + 420 x^{3} y - 35 x^{3} - 1600 x^{2} y^{3} + 2400 x^{2} y^{2} - 960 x^{2} y + 80 x^{2} + 1160 x y^{3} - 1740 x y^{2} + 696 x y - 58 x - 260 y^{3} + 390 y^{2} - 156 y + 13\right)\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{23}:\mathbf{v}\mapsto\displaystyle\int_{R}\left(\begin{array}{c}\displaystyle 0\\\displaystyle - 2 s_{0}\end{array}\right)\cdot\mathbf{v}\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).

\(\displaystyle \boldsymbol{\phi}_{23} = \left(\begin{array}{c}\displaystyle \frac{15 y \left(- 63700 x^{3} y^{3} + 163800 x^{3} y^{2} - 141960 x^{3} y + 41860 x^{3} + 95550 x^{2} y^{3} - 246480 x^{2} y^{2} + 214200 x^{2} y - 63270 x^{2} - 38220 x y^{3} + 99060 x y^{2} - 86436 x y + 25596 x + 3185 y^{3} - 8320 y^{2} + 7308 y - 2173\right)}{13}\\\displaystyle \frac{15 x \left(63700 x^{3} y^{3} - 95550 x^{3} y^{2} + 38220 x^{3} y - 4095 x^{3} - 134680 x^{2} y^{3} + 202800 x^{2} y^{2} - 81840 x^{2} y + 8862 x^{2} + 87360 x y^{3} - 132210 x y^{2} + 53964 x y - 5922 x - 16380 y^{3} + 24960 y^{2} - 10344 y + 1155\right)}{13}\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{24}:\mathbf{v}\mapsto\displaystyle\int_{R}\left(\begin{array}{c}\displaystyle s_{0}^{2}\\\displaystyle - 2 s_{0} s_{1}\end{array}\right)\cdot\mathbf{v}\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).

\(\displaystyle \boldsymbol{\phi}_{24} = \left(\begin{array}{c}\displaystyle \frac{180 y \left(63700 x^{3} y^{3} - 145600 x^{3} y^{2} + 105560 x^{3} y - 23660 x^{3} - 95550 x^{2} y^{3} + 218790 x^{2} y^{2} - 158940 x^{2} y + 35700 x^{2} + 38220 x y^{3} - 87750 x y^{2} + 63936 x y - 14406 x - 3185 y^{3} + 7345 y^{2} - 5378 y + 1218\right)}{13}\\\displaystyle \frac{180 x \left(- 63700 x^{3} y^{3} + 95550 x^{3} y^{2} - 38220 x^{3} y + 3185 x^{3} + 134680 x^{2} y^{3} - 202410 x^{2} y^{2} + 81240 x^{2} y - 6820 x^{2} - 87360 x y^{3} + 131625 x y^{2} - 53064 x y + 4497 x + 16380 y^{3} - 24765 y^{2} + 10044 y - 862\right)}{13}\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{25}:\mathbf{v}\mapsto\displaystyle\int_{R}\left(\begin{array}{c}\displaystyle 2 s_{0}^{2} s_{1}\\\displaystyle - 2 s_{0} s_{1}^{2}\end{array}\right)\cdot\mathbf{v}\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).

\(\displaystyle \boldsymbol{\phi}_{25} = \left(\begin{array}{c}\displaystyle 450 y \left(- 4900 x^{3} y^{3} + 10360 x^{3} y^{2} - 6720 x^{3} y + 1260 x^{3} + 7350 x^{2} y^{3} - 15552 x^{2} y^{2} + 10098 x^{2} y - 1896 x^{2} - 2940 x y^{3} + 6228 x y^{2} - 4050 x y + 762 x + 245 y^{3} - 520 y^{2} + 339 y - 64\right)\\\displaystyle 450 x \left(4900 x^{3} y^{3} - 7350 x^{3} y^{2} + 2940 x^{3} y - 245 x^{3} - 10360 x^{2} y^{3} + 15552 x^{2} y^{2} - 6228 x^{2} y + 520 x^{2} + 6720 x y^{3} - 10098 x y^{2} + 4050 x y - 339 x - 1260 y^{3} + 1896 y^{2} - 762 y + 64\right)\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{26}:\mathbf{v}\mapsto\displaystyle\int_{R}\left(\begin{array}{c}\displaystyle 3 s_{0}^{2} s_{1}^{2}\\\displaystyle - 2 s_{0} s_{1}^{3}\end{array}\right)\cdot\mathbf{v}\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).

\(\displaystyle \boldsymbol{\phi}_{26} = \left(\begin{array}{c}\displaystyle 14700 y \left(100 x^{3} y^{3} - 200 x^{3} y^{2} + 120 x^{3} y - 20 x^{3} - 150 x^{2} y^{3} + 300 x^{2} y^{2} - 180 x^{2} y + 30 x^{2} + 60 x y^{3} - 120 x y^{2} + 72 x y - 12 x - 5 y^{3} + 10 y^{2} - 6 y + 1\right)\\\displaystyle 2100 x \left(- 700 x^{3} y^{3} + 1050 x^{3} y^{2} - 420 x^{3} y + 35 x^{3} + 1480 x^{2} y^{3} - 2220 x^{2} y^{2} + 888 x^{2} y - 74 x^{2} - 960 x y^{3} + 1440 x y^{2} - 576 x y + 48 x + 180 y^{3} - 270 y^{2} + 108 y - 9\right)\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{27}:\mathbf{v}\mapsto\displaystyle\int_{R}\left(\begin{array}{c}\displaystyle 0\\\displaystyle - 3 s_{0}^{2}\end{array}\right)\cdot\mathbf{v}\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).

\(\displaystyle \boldsymbol{\phi}_{27} = \left(\begin{array}{c}\displaystyle 70 y \left(700 x^{3} y^{3} - 1800 x^{3} y^{2} + 1560 x^{3} y - 460 x^{3} - 1050 x^{2} y^{3} + 2700 x^{2} y^{2} - 2340 x^{2} y + 690 x^{2} + 420 x y^{3} - 1080 x y^{2} + 936 x y - 276 x - 35 y^{3} + 90 y^{2} - 78 y + 23\right)\\\displaystyle 70 x \left(- 700 x^{3} y^{3} + 1050 x^{3} y^{2} - 420 x^{3} y + 45 x^{3} + 1400 x^{2} y^{3} - 2100 x^{2} y^{2} + 840 x^{2} y - 90 x^{2} - 840 x y^{3} + 1260 x y^{2} - 504 x y + 54 x + 140 y^{3} - 210 y^{2} + 84 y - 9\right)\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{28}:\mathbf{v}\mapsto\displaystyle\int_{R}\left(\begin{array}{c}\displaystyle s_{0}^{3}\\\displaystyle - 3 s_{0}^{2} s_{1}\end{array}\right)\cdot\mathbf{v}\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).

\(\displaystyle \boldsymbol{\phi}_{28} = \left(\begin{array}{c}\displaystyle 840 y \left(- 700 x^{3} y^{3} + 1600 x^{3} y^{2} - 1160 x^{3} y + 260 x^{3} + 1050 x^{2} y^{3} - 2400 x^{2} y^{2} + 1740 x^{2} y - 390 x^{2} - 420 x y^{3} + 960 x y^{2} - 696 x y + 156 x + 35 y^{3} - 80 y^{2} + 58 y - 13\right)\\\displaystyle 5880 x \left(100 x^{3} y^{3} - 150 x^{3} y^{2} + 60 x^{3} y - 5 x^{3} - 200 x^{2} y^{3} + 300 x^{2} y^{2} - 120 x^{2} y + 10 x^{2} + 120 x y^{3} - 180 x y^{2} + 72 x y - 6 x - 20 y^{3} + 30 y^{2} - 12 y + 1\right)\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{29}:\mathbf{v}\mapsto\displaystyle\int_{R}\left(\begin{array}{c}\displaystyle 2 s_{0}^{3} s_{1}\\\displaystyle - 3 s_{0}^{2} s_{1}^{2}\end{array}\right)\cdot\mathbf{v}\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).

\(\displaystyle \boldsymbol{\phi}_{29} = \left(\begin{array}{c}\displaystyle 2100 y \left(700 x^{3} y^{3} - 1480 x^{3} y^{2} + 960 x^{3} y - 180 x^{3} - 1050 x^{2} y^{3} + 2220 x^{2} y^{2} - 1440 x^{2} y + 270 x^{2} + 420 x y^{3} - 888 x y^{2} + 576 x y - 108 x - 35 y^{3} + 74 y^{2} - 48 y + 9\right)\\\displaystyle 14700 x \left(- 100 x^{3} y^{3} + 150 x^{3} y^{2} - 60 x^{3} y + 5 x^{3} + 200 x^{2} y^{3} - 300 x^{2} y^{2} + 120 x^{2} y - 10 x^{2} - 120 x y^{3} + 180 x y^{2} - 72 x y + 6 x + 20 y^{3} - 30 y^{2} + 12 y - 1\right)\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{30}:\mathbf{v}\mapsto\displaystyle\int_{R}\left(\begin{array}{c}\displaystyle 3 s_{0}^{3} s_{1}^{2}\\\displaystyle - 3 s_{0}^{2} s_{1}^{3}\end{array}\right)\cdot\mathbf{v}\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).

\(\displaystyle \boldsymbol{\phi}_{30} = \left(\begin{array}{c}\displaystyle 9800 y \left(- 100 x^{3} y^{3} + 200 x^{3} y^{2} - 120 x^{3} y + 20 x^{3} + 150 x^{2} y^{3} - 300 x^{2} y^{2} + 180 x^{2} y - 30 x^{2} - 60 x y^{3} + 120 x y^{2} - 72 x y + 12 x + 5 y^{3} - 10 y^{2} + 6 y - 1\right)\\\displaystyle 9800 x \left(100 x^{3} y^{3} - 150 x^{3} y^{2} + 60 x^{3} y - 5 x^{3} - 200 x^{2} y^{3} + 300 x^{2} y^{2} - 120 x^{2} y + 10 x^{2} + 120 x y^{3} - 180 x y^{2} + 72 x y - 6 x - 20 y^{3} + 30 y^{2} - 12 y + 1\right)\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{31}:\mathbf{v}\mapsto\displaystyle\int_{R}\left(\begin{array}{c}\displaystyle s_{1} \left(- 2 s_{0} s_{1} + 2 s_{0} + s_{1} - 1\right)\\\displaystyle s_{0} \left(- 2 s_{0} s_{1} + s_{0} + 2 s_{1} - 1\right)\end{array}\right)\cdot\mathbf{v}\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).

\(\displaystyle \boldsymbol{\phi}_{31} = \left(\begin{array}{c}\displaystyle \frac{60 y \left(- 1365 x^{2} y^{2} + 2205 x^{2} y - 840 x^{2} + 1470 x y^{2} - 2382 x y + 912 x - 280 y^{2} + 456 y - 176\right)}{13}\\\displaystyle \frac{60 x \left(- 1365 x^{2} y^{2} + 1470 x^{2} y - 280 x^{2} + 2205 x y^{2} - 2382 x y + 456 x - 840 y^{2} + 912 y - 176\right)}{13}\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{32}:\mathbf{v}\mapsto\displaystyle\int_{R}\left(\begin{array}{c}\displaystyle s_{1}^{2} \left(- 2 s_{0} s_{1} + 2 s_{0} + s_{1} - 1\right)\\\displaystyle s_{0} s_{1} \left(- 3 s_{0} s_{1} + 2 s_{0} + 3 s_{1} - 2\right)\end{array}\right)\cdot\mathbf{v}\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).

\(\displaystyle \boldsymbol{\phi}_{32} = \left(\begin{array}{c}\displaystyle \frac{2100 y \left(78 x^{2} y^{2} - 117 x^{2} y + 39 x^{2} - 84 x y^{2} + 126 x y - 42 x + 16 y^{2} - 24 y + 8\right)}{13}\\\displaystyle \frac{2100 x \left(78 x^{2} y^{2} - 78 x^{2} y + 13 x^{2} - 126 x y^{2} + 126 x y - 21 x + 48 y^{2} - 48 y + 8\right)}{13}\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{33}:\mathbf{v}\mapsto\displaystyle\int_{R}\left(\begin{array}{c}\displaystyle s_{0} s_{1} \left(- 3 s_{0} s_{1} + 3 s_{0} + 2 s_{1} - 2\right)\\\displaystyle s_{0}^{2} \left(- 2 s_{0} s_{1} + s_{0} + 2 s_{1} - 1\right)\end{array}\right)\cdot\mathbf{v}\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).

\(\displaystyle \boldsymbol{\phi}_{33} = \left(\begin{array}{c}\displaystyle \frac{2100 y \left(78 x^{2} y^{2} - 126 x^{2} y + 48 x^{2} - 78 x y^{2} + 126 x y - 48 x + 13 y^{2} - 21 y + 8\right)}{13}\\\displaystyle \frac{2100 x \left(78 x^{2} y^{2} - 84 x^{2} y + 16 x^{2} - 117 x y^{2} + 126 x y - 24 x + 39 y^{2} - 42 y + 8\right)}{13}\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{34}:\mathbf{v}\mapsto\displaystyle\int_{R}\left(\begin{array}{c}\displaystyle s_{0} s_{1}^{2} \left(- 3 s_{0} s_{1} + 3 s_{0} + 2 s_{1} - 2\right)\\\displaystyle s_{0}^{2} s_{1} \left(- 3 s_{0} s_{1} + 2 s_{0} + 3 s_{1} - 2\right)\end{array}\right)\cdot\mathbf{v}\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).

\(\displaystyle \boldsymbol{\phi}_{34} = \left(\begin{array}{c}\displaystyle 2100 y \left(- 12 x^{2} y^{2} + 18 x^{2} y - 6 x^{2} + 12 x y^{2} - 18 x y + 6 x - 2 y^{2} + 3 y - 1\right)\\\displaystyle 2100 x \left(- 12 x^{2} y^{2} + 12 x^{2} y - 2 x^{2} + 18 x y^{2} - 18 x y + 3 x - 6 y^{2} + 6 y - 1\right)\end{array}\right)\)

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: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{3 x z \left(z - 1\right)}{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle \frac{3 x y \left(1 - y\right)}{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{3 x z \left(- 2 y z + 2 y + z - 1\right)}{2}\\\displaystyle \frac{3 x y \left(2 y z - y - 2 z + 1\right)}{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{3 z \left(- x z + x + z - 1\right)}{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle \frac{3 y \left(x y - x - y + 1\right)}{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{3 z \left(2 x y z - 2 x y - x z + x - 2 y z + 2 y + z - 1\right)}{2}\\\displaystyle \frac{3 y \left(- 2 x y z + x y + 2 x z - x + 2 y z - y - 2 z + 1\right)}{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle \frac{3 y z \left(1 - z\right)}{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle \frac{3 x y \left(x - 1\right)}{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle \frac{3 y z \left(2 x z - 2 x - z + 1\right)}{2}\\\displaystyle 0\\\displaystyle \frac{3 x y \left(- 2 x z + x + 2 z - 1\right)}{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle \frac{3 z \left(y z - y - z + 1\right)}{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle \frac{3 x \left(- x y + x + y - 1\right)}{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle \frac{3 z \left(- 2 x y z + 2 x y + 2 x z - 2 x + y z - y - z + 1\right)}{2}\\\displaystyle 0\\\displaystyle \frac{3 x \left(2 x y z - x y - 2 x z + x - 2 y z + y + 2 z - 1\right)}{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle \frac{3 y z \left(y - 1\right)}{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{3 x z \left(1 - x\right)}{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle \frac{3 y z \left(- 2 x y + 2 x + y - 1\right)}{2}\\\displaystyle \frac{3 x z \left(2 x y - x - 2 y + 1\right)}{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle \frac{3 y \left(- y z + y + z - 1\right)}{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{3 x \left(x z - x - z + 1\right)}{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle \frac{3 y \left(2 x y z - 2 x y - 2 x z + 2 x - y z + y + z - 1\right)}{2}\\\displaystyle \frac{3 x \left(- 2 x y z + 2 x y + x z - x + 2 y z - 2 y - z + 1\right)}{2}\\\displaystyle 0\end{array}\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{41}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{0}\)
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{t}}_{0}\) is the tangent to edge 0;
and \(s_{0}\) is a parametrisation of \(e_{0}\).

\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 9 x y^{2} z - 9 x y^{2} + 9 x y z^{2} - 24 x y z + 15 x y - 9 x z^{2} + 15 x z - 6 x - \frac{15 y^{2} z}{2} + \frac{15 y^{2}}{2} - \frac{15 y z^{2}}{2} + 19 y z - \frac{23 y}{2} + \frac{15 z^{2}}{2} - \frac{23 z}{2} + 4\\\displaystyle \frac{9 x \left(- 2 x y z + 2 x y + x z - x + 2 y z - 2 y - z + 1\right)}{2}\\\displaystyle \frac{9 x \left(- 2 x y z + x y + 2 x z - x + 2 y z - y - 2 z + 1\right)}{2}\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle - 9 x y^{2} z + 9 x y^{2} - 9 x y z^{2} + 24 x y z - 15 x y + 9 x z^{2} - 15 x z + 6 x + \frac{3 y^{2} z}{2} - \frac{3 y^{2}}{2} + \frac{3 y z^{2}}{2} - 5 y z + \frac{7 y}{2} - \frac{3 z^{2}}{2} + \frac{7 z}{2} - 2\\\displaystyle \frac{9 x \left(2 x y z - 2 x y - x z + x - 2 y z + 2 y + z - 1\right)}{2}\\\displaystyle \frac{9 x \left(2 x y z - x y - 2 x z + x - 2 y z + y + 2 z - 1\right)}{2}\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle \frac{9 y \left(- 2 x y z + 2 x y + 2 x z - 2 x + y z - y - z + 1\right)}{2}\\\displaystyle 9 x^{2} y z - 9 x^{2} y - \frac{15 x^{2} z}{2} + \frac{15 x^{2}}{2} + 9 x y z^{2} - 24 x y z + 15 x y - \frac{15 x z^{2}}{2} + 19 x z - \frac{23 x}{2} - 9 y z^{2} + 15 y z - 6 y + \frac{15 z^{2}}{2} - \frac{23 z}{2} + 4\\\displaystyle \frac{9 y \left(- 2 x y z + x y + 2 x z - x + 2 y z - y - 2 z + 1\right)}{2}\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle \frac{9 y \left(2 x y z - 2 x y - 2 x z + 2 x - y z + y + z - 1\right)}{2}\\\displaystyle - 9 x^{2} y z + 9 x^{2} y + \frac{3 x^{2} z}{2} - \frac{3 x^{2}}{2} - 9 x y z^{2} + 24 x y z - 15 x y + \frac{3 x z^{2}}{2} - 5 x z + \frac{7 x}{2} + 9 y z^{2} - 15 y z + 6 y - \frac{3 z^{2}}{2} + \frac{7 z}{2} - 2\\\displaystyle \frac{9 y \left(2 x y z - x y - 2 x z + x - 2 y z + y + 2 z - 1\right)}{2}\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle \frac{9 z \left(- 2 x y z + 2 x y + 2 x z - 2 x + y z - y - z + 1\right)}{2}\\\displaystyle \frac{9 z \left(- 2 x y z + 2 x y + x z - x + 2 y z - 2 y - z + 1\right)}{2}\\\displaystyle 9 x^{2} y z - \frac{15 x^{2} y}{2} - 9 x^{2} z + \frac{15 x^{2}}{2} + 9 x y^{2} z - \frac{15 x y^{2}}{2} - 24 x y z + 19 x y + 15 x z - \frac{23 x}{2} - 9 y^{2} z + \frac{15 y^{2}}{2} + 15 y z - \frac{23 y}{2} - 6 z + 4\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle \frac{9 z \left(2 x y z - 2 x y - 2 x z + 2 x - y z + y + z - 1\right)}{2}\\\displaystyle \frac{9 z \left(2 x y z - 2 x y - x z + x - 2 y z + 2 y + z - 1\right)}{2}\\\displaystyle - 9 x^{2} y z + \frac{3 x^{2} y}{2} + 9 x^{2} z - \frac{3 x^{2}}{2} - 9 x y^{2} z + \frac{3 x y^{2}}{2} + 24 x y z - 5 x y - 15 x z + \frac{7 x}{2} + 9 y^{2} z - \frac{3 y^{2}}{2} - 15 y z + \frac{7 y}{2} + 6 z - 2\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle \frac{9 y \left(- 2 x y z + 2 x y + 2 x z - 2 x + y z - y - z + 1\right)}{2}\\\displaystyle \frac{x \left(18 x y z - 18 x y - 15 x z + 15 x - 18 y z^{2} + 12 y z + 6 y + 15 z^{2} - 8 z - 7\right)}{2}\\\displaystyle \frac{9 x y \left(2 y z - y - 2 z + 1\right)}{2}\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle \frac{9 y \left(2 x y z - 2 x y - 2 x z + 2 x - y z + y + z - 1\right)}{2}\\\displaystyle \frac{x \left(- 18 x y z + 18 x y + 3 x z - 3 x + 18 y z^{2} - 12 y z - 6 y - 3 z^{2} + 4 z - 1\right)}{2}\\\displaystyle \frac{9 x y \left(- 2 y z + y + 2 z - 1\right)}{2}\end{array}\right)\)

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{4}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{4}\)
where \(e_{4}\) is the 4th edge;
\(\hat{\boldsymbol{t}}_{4}\) is the tangent to edge 4;
and \(s_{0}\) is a parametrisation of \(e_{4}\).

\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle \frac{9 z \left(- 2 x y z + 2 x y + 2 x z - 2 x + y z - y - z + 1\right)}{2}\\\displaystyle \frac{9 x z \left(2 y z - 2 y - z + 1\right)}{2}\\\displaystyle \frac{x \left(18 x y z - 15 x y - 18 x z + 15 x - 18 y^{2} z + 15 y^{2} + 12 y z - 8 y + 6 z - 7\right)}{2}\end{array}\right)\)

This DOF is associated with edge 4 of the reference element.
\(\displaystyle l_{9}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{4}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{4}\)
where \(e_{4}\) is the 4th edge;
\(\hat{\boldsymbol{t}}_{4}\) is the tangent to edge 4;
and \(s_{0}\) is a parametrisation of \(e_{4}\).

\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle \frac{9 z \left(2 x y z - 2 x y - 2 x z + 2 x - y z + y + z - 1\right)}{2}\\\displaystyle \frac{9 x z \left(- 2 y z + 2 y + z - 1\right)}{2}\\\displaystyle \frac{x \left(- 18 x y z + 3 x y + 18 x z - 3 x + 18 y^{2} z - 3 y^{2} - 12 y z + 4 y - 6 z - 1\right)}{2}\end{array}\right)\)

This DOF is associated with edge 4 of the reference element.
\(\displaystyle l_{10}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{5}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{5}\)
where \(e_{5}\) is the 5th edge;
\(\hat{\boldsymbol{t}}_{5}\) is the tangent to edge 5;
and \(s_{0}\) is a parametrisation of \(e_{5}\).

\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle \frac{y \left(18 x y z - 18 x y - 18 x z^{2} + 12 x z + 6 x - 15 y z + 15 y + 15 z^{2} - 8 z - 7\right)}{2}\\\displaystyle \frac{9 x \left(- 2 x y z + 2 x y + x z - x + 2 y z - 2 y - z + 1\right)}{2}\\\displaystyle \frac{9 x y \left(2 x z - x - 2 z + 1\right)}{2}\end{array}\right)\)

This DOF is associated with edge 5 of the reference element.
\(\displaystyle l_{11}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{5}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{5}\)
where \(e_{5}\) is the 5th edge;
\(\hat{\boldsymbol{t}}_{5}\) is the tangent to edge 5;
and \(s_{0}\) is a parametrisation of \(e_{5}\).

\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle \frac{y \left(- 18 x y z + 18 x y + 18 x z^{2} - 12 x z - 6 x + 3 y z - 3 y - 3 z^{2} + 4 z - 1\right)}{2}\\\displaystyle \frac{9 x \left(2 x y z - 2 x y - x z + x - 2 y z + 2 y + z - 1\right)}{2}\\\displaystyle \frac{9 x y \left(- 2 x z + x + 2 z - 1\right)}{2}\end{array}\right)\)

This DOF is associated with edge 5 of the reference element.
\(\displaystyle l_{12}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{6}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{6}\)
where \(e_{6}\) is the 6th edge;
\(\hat{\boldsymbol{t}}_{6}\) is the tangent to edge 6;
and \(s_{0}\) is a parametrisation of \(e_{6}\).

\(\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle \frac{9 y z \left(2 x z - 2 x - z + 1\right)}{2}\\\displaystyle \frac{9 z \left(- 2 x y z + 2 x y + x z - x + 2 y z - 2 y - z + 1\right)}{2}\\\displaystyle \frac{y \left(- 18 x^{2} z + 15 x^{2} + 18 x y z - 15 x y + 12 x z - 8 x - 18 y z + 15 y + 6 z - 7\right)}{2}\end{array}\right)\)

This DOF is associated with edge 6 of the reference element.
\(\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{6}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{6}\)
where \(e_{6}\) is the 6th edge;
\(\hat{\boldsymbol{t}}_{6}\) is the tangent to edge 6;
and \(s_{0}\) is a parametrisation of \(e_{6}\).

\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle \frac{9 y z \left(- 2 x z + 2 x + z - 1\right)}{2}\\\displaystyle \frac{9 z \left(2 x y z - 2 x y - x z + x - 2 y z + 2 y + z - 1\right)}{2}\\\displaystyle \frac{y \left(18 x^{2} z - 3 x^{2} - 18 x y z + 3 x y - 12 x z + 4 x + 18 y z - 3 y - 6 z - 1\right)}{2}\end{array}\right)\)

This DOF is associated with edge 6 of the reference element.
\(\displaystyle l_{14}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{7}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{7}\)
where \(e_{7}\) is the 7th edge;
\(\hat{\boldsymbol{t}}_{7}\) is the tangent to edge 7;
and \(s_{0}\) is a parametrisation of \(e_{7}\).

\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle \frac{9 y z \left(2 x z - 2 x - z + 1\right)}{2}\\\displaystyle \frac{9 x z \left(2 y z - 2 y - z + 1\right)}{2}\\\displaystyle \frac{x y \left(- 18 x z + 15 x - 18 y z + 15 y + 24 z - 22\right)}{2}\end{array}\right)\)

This DOF is associated with edge 7 of the reference element.
\(\displaystyle l_{15}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{7}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{7}\)
where \(e_{7}\) is the 7th edge;
\(\hat{\boldsymbol{t}}_{7}\) is the tangent to edge 7;
and \(s_{0}\) is a parametrisation of \(e_{7}\).

\(\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle \frac{9 y z \left(- 2 x z + 2 x + z - 1\right)}{2}\\\displaystyle \frac{9 x z \left(- 2 y z + 2 y + z - 1\right)}{2}\\\displaystyle \frac{x y \left(18 x z - 3 x + 18 y z - 3 y - 24 z + 2\right)}{2}\end{array}\right)\)

This DOF is associated with edge 7 of the reference element.
\(\displaystyle l_{16}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{8}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{8}\)
where \(e_{8}\) is the 8th edge;
\(\hat{\boldsymbol{t}}_{8}\) is the tangent to edge 8;
and \(s_{0}\) is a parametrisation of \(e_{8}\).

\(\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle \frac{z \left(- 18 x y^{2} + 18 x y z + 12 x y - 18 x z + 6 x + 15 y^{2} - 15 y z - 8 y + 15 z - 7\right)}{2}\\\displaystyle \frac{9 x z \left(2 x y - x - 2 y + 1\right)}{2}\\\displaystyle \frac{9 x \left(- 2 x y z + x y + 2 x z - x + 2 y z - y - 2 z + 1\right)}{2}\end{array}\right)\)

This DOF is associated with edge 8 of the reference element.
\(\displaystyle l_{17}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{8}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{8}\)
where \(e_{8}\) is the 8th edge;
\(\hat{\boldsymbol{t}}_{8}\) is the tangent to edge 8;
and \(s_{0}\) is a parametrisation of \(e_{8}\).

\(\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle \frac{z \left(18 x y^{2} - 18 x y z - 12 x y + 18 x z - 6 x - 3 y^{2} + 3 y z + 4 y - 3 z - 1\right)}{2}\\\displaystyle \frac{9 x z \left(- 2 x y + x + 2 y - 1\right)}{2}\\\displaystyle \frac{9 x \left(2 x y z - x y - 2 x z + x - 2 y z + y + 2 z - 1\right)}{2}\end{array}\right)\)

This DOF is associated with edge 8 of the reference element.
\(\displaystyle l_{18}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{9}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{9}\)
where \(e_{9}\) is the 9th edge;
\(\hat{\boldsymbol{t}}_{9}\) is the tangent to edge 9;
and \(s_{0}\) is a parametrisation of \(e_{9}\).

\(\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle \frac{9 y z \left(2 x y - 2 x - y + 1\right)}{2}\\\displaystyle \frac{z \left(- 18 x^{2} y + 15 x^{2} + 18 x y z + 12 x y - 15 x z - 8 x - 18 y z + 6 y + 15 z - 7\right)}{2}\\\displaystyle \frac{9 y \left(- 2 x y z + x y + 2 x z - x + 2 y z - y - 2 z + 1\right)}{2}\end{array}\right)\)

This DOF is associated with edge 9 of the reference element.
\(\displaystyle l_{19}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{9}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{9}\)
where \(e_{9}\) is the 9th edge;
\(\hat{\boldsymbol{t}}_{9}\) is the tangent to edge 9;
and \(s_{0}\) is a parametrisation of \(e_{9}\).

\(\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle \frac{9 y z \left(- 2 x y + 2 x + y - 1\right)}{2}\\\displaystyle \frac{z \left(18 x^{2} y - 3 x^{2} - 18 x y z - 12 x y + 3 x z + 4 x + 18 y z - 6 y - 3 z - 1\right)}{2}\\\displaystyle \frac{9 y \left(2 x y z - x y - 2 x z + x - 2 y z + y + 2 z - 1\right)}{2}\end{array}\right)\)

This DOF is associated with edge 9 of the reference element.
\(\displaystyle l_{20}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{10}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{10}\)
where \(e_{10}\) is the 10th edge;
\(\hat{\boldsymbol{t}}_{10}\) is the tangent to edge 10;
and \(s_{0}\) is a parametrisation of \(e_{10}\).

\(\displaystyle \boldsymbol{\phi}_{20} = \left(\begin{array}{c}\displaystyle \frac{9 y z \left(2 x y - 2 x - y + 1\right)}{2}\\\displaystyle \frac{x z \left(- 18 x y + 15 x - 18 y z + 24 y + 15 z - 22\right)}{2}\\\displaystyle \frac{9 x y \left(2 y z - y - 2 z + 1\right)}{2}\end{array}\right)\)

This DOF is associated with edge 10 of the reference element.
\(\displaystyle l_{21}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{10}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{10}\)
where \(e_{10}\) is the 10th edge;
\(\hat{\boldsymbol{t}}_{10}\) is the tangent to edge 10;
and \(s_{0}\) is a parametrisation of \(e_{10}\).

\(\displaystyle \boldsymbol{\phi}_{21} = \left(\begin{array}{c}\displaystyle \frac{9 y z \left(- 2 x y + 2 x + y - 1\right)}{2}\\\displaystyle \frac{x z \left(18 x y - 3 x + 18 y z - 24 y - 3 z + 2\right)}{2}\\\displaystyle \frac{9 x y \left(- 2 y z + y + 2 z - 1\right)}{2}\end{array}\right)\)

This DOF is associated with edge 10 of the reference element.
\(\displaystyle l_{22}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{11}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{11}\)
where \(e_{11}\) is the 11th edge;
\(\hat{\boldsymbol{t}}_{11}\) is the tangent to edge 11;
and \(s_{0}\) is a parametrisation of \(e_{11}\).

\(\displaystyle \boldsymbol{\phi}_{22} = \left(\begin{array}{c}\displaystyle \frac{y z \left(- 18 x y - 18 x z + 24 x + 15 y + 15 z - 22\right)}{2}\\\displaystyle \frac{9 x z \left(2 x y - x - 2 y + 1\right)}{2}\\\displaystyle \frac{9 x y \left(2 x z - x - 2 z + 1\right)}{2}\end{array}\right)\)

This DOF is associated with edge 11 of the reference element.
\(\displaystyle l_{23}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{11}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{11}\)
where \(e_{11}\) is the 11th edge;
\(\hat{\boldsymbol{t}}_{11}\) is the tangent to edge 11;
and \(s_{0}\) is a parametrisation of \(e_{11}\).

\(\displaystyle \boldsymbol{\phi}_{23} = \left(\begin{array}{c}\displaystyle \frac{y z \left(18 x y + 18 x z - 24 x - 3 y - 3 z + 2\right)}{2}\\\displaystyle \frac{9 x z \left(- 2 x y + x + 2 y - 1\right)}{2}\\\displaystyle \frac{9 x y \left(- 2 x z + x + 2 z - 1\right)}{2}\end{array}\right)\)

This DOF is associated with edge 11 of the reference element.
\(\displaystyle l_{24}:\mathbf{v}\mapsto\displaystyle\int_{f_{0}}\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(f_{0}\) is the 0th face.

\(\displaystyle \boldsymbol{\phi}_{24} = \left(\begin{array}{c}\displaystyle 3 y \left(- 6 x y z + 6 x y + 6 x z - 6 x + 5 y z - 5 y - 5 z + 5\right)\\\displaystyle 9 x \left(2 x y z - 2 x y - x z + x - 2 y z + 2 y + z - 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{25}:\mathbf{v}\mapsto\displaystyle\int_{f_{0}}\left(\begin{array}{c}\displaystyle 0\\\displaystyle -1\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(f_{0}\) is the 0th face.

\(\displaystyle \boldsymbol{\phi}_{25} = \left(\begin{array}{c}\displaystyle 9 y \left(- 2 x y z + 2 x y + 2 x z - 2 x + y z - y - z + 1\right)\\\displaystyle 3 x \left(6 x y z - 6 x y - 5 x z + 5 x - 6 y z + 6 y + 5 z - 5\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{26}:\mathbf{v}\mapsto\displaystyle\int_{f_{0}}\left(\begin{array}{c}\displaystyle s_{0}\\\displaystyle - s_{1}\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(f_{0}\) is the 0th face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{0}\).

\(\displaystyle \boldsymbol{\phi}_{26} = \left(\begin{array}{c}\displaystyle 18 y \left(2 x y z - 2 x y - 2 x z + 2 x - y z + y + z - 1\right)\\\displaystyle 18 x \left(- 2 x y z + 2 x y + x z - x + 2 y z - 2 y - z + 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{27}:\mathbf{v}\mapsto\displaystyle\int_{f_{1}}\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(f_{1}\) is the 1st face.

\(\displaystyle \boldsymbol{\phi}_{27} = \left(\begin{array}{c}\displaystyle 3 z \left(- 6 x y z + 6 x y + 6 x z - 6 x + 5 y z - 5 y - 5 z + 5\right)\\\displaystyle 0\\\displaystyle 9 x \left(2 x y z - x y - 2 x z + x - 2 y z + y + 2 z - 1\right)\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{28}:\mathbf{v}\mapsto\displaystyle\int_{f_{1}}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle -1\end{array}\right)\cdot\mathbf{v}\)
where \(f_{1}\) is the 1st face.

\(\displaystyle \boldsymbol{\phi}_{28} = \left(\begin{array}{c}\displaystyle 9 z \left(- 2 x y z + 2 x y + 2 x z - 2 x + y z - y - z + 1\right)\\\displaystyle 0\\\displaystyle 3 x \left(6 x y z - 5 x y - 6 x z + 5 x - 6 y z + 5 y + 6 z - 5\right)\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{29}:\mathbf{v}\mapsto\displaystyle\int_{f_{1}}\left(\begin{array}{c}\displaystyle s_{0}\\\displaystyle 0\\\displaystyle - s_{1}\end{array}\right)\cdot\mathbf{v}\)
where \(f_{1}\) is the 1st face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{1}\).

\(\displaystyle \boldsymbol{\phi}_{29} = \left(\begin{array}{c}\displaystyle 18 z \left(2 x y z - 2 x y - 2 x z + 2 x - y z + y + z - 1\right)\\\displaystyle 0\\\displaystyle 18 x \left(- 2 x y z + x y + 2 x z - x + 2 y z - y - 2 z + 1\right)\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{30}:\mathbf{v}\mapsto\displaystyle\int_{f_{2}}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(f_{2}\) is the 2nd face.

\(\displaystyle \boldsymbol{\phi}_{30} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 3 z \left(- 6 x y z + 6 x y + 5 x z - 5 x + 6 y z - 6 y - 5 z + 5\right)\\\displaystyle 9 y \left(2 x y z - x y - 2 x z + x - 2 y z + y + 2 z - 1\right)\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{31}:\mathbf{v}\mapsto\displaystyle\int_{f_{2}}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle -1\end{array}\right)\cdot\mathbf{v}\)
where \(f_{2}\) is the 2nd face.

\(\displaystyle \boldsymbol{\phi}_{31} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 9 z \left(- 2 x y z + 2 x y + x z - x + 2 y z - 2 y - z + 1\right)\\\displaystyle 3 y \left(6 x y z - 5 x y - 6 x z + 5 x - 6 y z + 5 y + 6 z - 5\right)\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{32}:\mathbf{v}\mapsto\displaystyle\int_{f_{2}}\left(\begin{array}{c}\displaystyle 0\\\displaystyle s_{0}\\\displaystyle - s_{1}\end{array}\right)\cdot\mathbf{v}\)
where \(f_{2}\) is the 2nd face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{2}\).

\(\displaystyle \boldsymbol{\phi}_{32} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 18 z \left(2 x y z - 2 x y - x z + x - 2 y z + 2 y + z - 1\right)\\\displaystyle 18 y \left(- 2 x y z + x y + 2 x z - x + 2 y z - y - 2 z + 1\right)\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{33}:\mathbf{v}\mapsto\displaystyle\int_{f_{3}}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(f_{3}\) is the 3th face.

\(\displaystyle \boldsymbol{\phi}_{33} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 3 x z \left(6 y z - 6 y - 5 z + 5\right)\\\displaystyle 9 x y \left(- 2 y z + y + 2 z - 1\right)\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{34}:\mathbf{v}\mapsto\displaystyle\int_{f_{3}}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle -1\end{array}\right)\cdot\mathbf{v}\)
where \(f_{3}\) is the 3th face.

\(\displaystyle \boldsymbol{\phi}_{34} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 9 x z \left(2 y z - 2 y - z + 1\right)\\\displaystyle 3 x y \left(- 6 y z + 5 y + 6 z - 5\right)\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{35}:\mathbf{v}\mapsto\displaystyle\int_{f_{3}}\left(\begin{array}{c}\displaystyle 0\\\displaystyle s_{0}\\\displaystyle - s_{1}\end{array}\right)\cdot\mathbf{v}\)
where \(f_{3}\) is the 3th face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{3}\).

\(\displaystyle \boldsymbol{\phi}_{35} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 18 x z \left(- 2 y z + 2 y + z - 1\right)\\\displaystyle 18 x y \left(2 y z - y - 2 z + 1\right)\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{36}:\mathbf{v}\mapsto\displaystyle\int_{f_{4}}\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(f_{4}\) is the 4th face.

\(\displaystyle \boldsymbol{\phi}_{36} = \left(\begin{array}{c}\displaystyle 3 y z \left(6 x z - 6 x - 5 z + 5\right)\\\displaystyle 0\\\displaystyle 9 x y \left(- 2 x z + x + 2 z - 1\right)\end{array}\right)\)

This DOF is associated with face 4 of the reference element.
\(\displaystyle l_{37}:\mathbf{v}\mapsto\displaystyle\int_{f_{4}}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle -1\end{array}\right)\cdot\mathbf{v}\)
where \(f_{4}\) is the 4th face.

\(\displaystyle \boldsymbol{\phi}_{37} = \left(\begin{array}{c}\displaystyle 9 y z \left(2 x z - 2 x - z + 1\right)\\\displaystyle 0\\\displaystyle 3 x y \left(- 6 x z + 5 x + 6 z - 5\right)\end{array}\right)\)

This DOF is associated with face 4 of the reference element.
\(\displaystyle l_{38}:\mathbf{v}\mapsto\displaystyle\int_{f_{4}}\left(\begin{array}{c}\displaystyle s_{0}\\\displaystyle 0\\\displaystyle - s_{1}\end{array}\right)\cdot\mathbf{v}\)
where \(f_{4}\) is the 4th face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{4}\).

\(\displaystyle \boldsymbol{\phi}_{38} = \left(\begin{array}{c}\displaystyle 18 y z \left(- 2 x z + 2 x + z - 1\right)\\\displaystyle 0\\\displaystyle 18 x y \left(2 x z - x - 2 z + 1\right)\end{array}\right)\)

This DOF is associated with face 4 of the reference element.
\(\displaystyle l_{39}:\mathbf{v}\mapsto\displaystyle\int_{f_{5}}\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(f_{5}\) is the 5th face.

\(\displaystyle \boldsymbol{\phi}_{39} = \left(\begin{array}{c}\displaystyle 3 y z \left(6 x y - 6 x - 5 y + 5\right)\\\displaystyle 9 x z \left(- 2 x y + x + 2 y - 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 5 of the reference element.
\(\displaystyle l_{40}:\mathbf{v}\mapsto\displaystyle\int_{f_{5}}\left(\begin{array}{c}\displaystyle 0\\\displaystyle -1\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(f_{5}\) is the 5th face.

\(\displaystyle \boldsymbol{\phi}_{40} = \left(\begin{array}{c}\displaystyle 9 y z \left(2 x y - 2 x - y + 1\right)\\\displaystyle 3 x z \left(- 6 x y + 5 x + 6 y - 5\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 5 of the reference element.
\(\displaystyle l_{41}:\mathbf{v}\mapsto\displaystyle\int_{f_{5}}\left(\begin{array}{c}\displaystyle s_{0}\\\displaystyle - s_{1}\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(f_{5}\) is the 5th face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{5}\).

\(\displaystyle \boldsymbol{\phi}_{41} = \left(\begin{array}{c}\displaystyle 18 y z \left(- 2 x y + 2 x + y - 1\right)\\\displaystyle 18 x z \left(2 x y - x - 2 y + 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 5 of the reference element.

References

DefElement stats

Element added24 October 2021
Element last updated24 October 2021