an encyclopedia of finite element definitions

# Serendipity H(curl)

 Alternative names Brezzi–Douglas–Marini cubical H(curl) (quadrilateral), Arnold–Awanou H(curl) (hexahedron) Exterior calculus names $$\mathcal{S}_{k}\Lambda^{1}(\square_d)$$ Cockburn–fu names $$\left[S_{1,k}^\square\right]_{1}$$ Abbreviated names BDMce (quadrilateral), AAe (hexahedron) Orders $$1\leqslant k$$ Reference elements quadrilateral, hexahedron Polynomial set $$\mathcal{P}_{k}^d \oplus \mathcal{Z}^{(5)}_{k}$$ (quadrilateral) $$\mathcal{P}_{k}^d \oplus \mathcal{A}_{k-1} \oplus \mathcal{Z}^{(6)}_{k}$$ (hexahedron) ↓ Show polynomial set definitions ↓ DOFs On each edge: tangent integral moments with an order $$k$$ dPc space On each face: integral moments with an order $$k-2$$ vector dPc space On each volume: integral moments with an order $$k-4$$ vector dPc space Number of DOFs quadrilateral: $$k^2+3k+4$$ (A014206)hexahedron: $$\begin{cases}6(k^2+k+2)&k=1,2,3\\k(k+1)(k-1)/2 + 3k^2 + 12k + 9&k > 3\end{cases}$$ Categories Vector-valued elements, H(curl) conforming elements

## Implementations

 Symfem "Scurl"↓ Show Symfem examples ↓ UFL "BDMCE" (quadrilateral)"AAE" (hexahedron)↓ Show UFL examples ↓

## Examples

order 1
order 2
hexahedron
order 1
hexahedron
order 2
• $$R$$ is the reference quadrilateral. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 2 x y\\\displaystyle - x^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle - 2 x y\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{7}\}$$
• 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 $$e_{0}$$.

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

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 $$e_{0}$$.

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

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 $$e_{1}$$.

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

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 $$e_{1}$$.

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

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 $$e_{2}$$.

$$\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 3 y \left(y - 1\right)\\\displaystyle 2 x \left(2 - 3 y\right)\end{array}\right)$$

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 $$e_{2}$$.

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

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 $$e_{3}$$.

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

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

$$\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 2 y \left(3 x - 1\right)\\\displaystyle 3 x \left(1 - x\right)\end{array}\right)$$

This DOF is associated with edge 3 of the reference element.
• $$R$$ is the reference quadrilateral. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 3 x^{2} y\\\displaystyle - x^{3}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y^{3}\\\displaystyle - 3 x y^{2}\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{13}\}$$
• 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 $$e_{0}$$.

$$\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle - 30 x^{2} y + 30 x^{2} + 36 x y - 36 x + 3 y^{2} - 12 y + 9\\\displaystyle 5 x \left(2 x^{2} - 3 x + 1\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle - 30 x^{2} y + 30 x^{2} + 24 x y - 24 x + 3 y^{2} - 6 y + 3\\\displaystyle 5 x \left(2 x^{2} - 3 x + 1\right)\end{array}\right)$$

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

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

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 $$e_{1}$$.

$$\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 5 y \left(2 y^{2} - 3 y + 1\right)\\\displaystyle 3 x^{2} - 30 x y^{2} + 36 x y - 12 x + 30 y^{2} - 36 y + 9\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 5 y \left(2 y^{2} - 3 y + 1\right)\\\displaystyle 3 x^{2} - 30 x y^{2} + 24 x y - 6 x + 30 y^{2} - 24 y + 3\end{array}\right)$$

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

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

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 $$e_{2}$$.

$$\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 5 y \left(- 2 y^{2} + 3 y - 1\right)\\\displaystyle 3 x \left(x + 10 y^{2} - 12 y + 2\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 5 y \left(- 2 y^{2} + 3 y - 1\right)\\\displaystyle 3 x \left(x + 10 y^{2} - 8 y\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle \frac{5 y \left(2 y^{2} - 3 y + 1\right)}{2}\\\displaystyle \frac{3 x \left(2 x - 10 y^{2} + 10 y - 3\right)}{2}\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 3 y \left(10 x^{2} - 12 x + y + 2\right)\\\displaystyle 5 x \left(- 2 x^{2} + 3 x - 1\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 3 y \left(10 x^{2} - 8 x + y\right)\\\displaystyle 5 x \left(- 2 x^{2} + 3 x - 1\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle \frac{3 y \left(- 10 x^{2} + 10 x + 2 y - 3\right)}{2}\\\displaystyle \frac{5 x \left(2 x^{2} - 3 x + 1\right)}{2}\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 6 y \left(1 - y\right)\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 6 x \left(1 - x\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
• $$R$$ is the reference hexahedron. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x z\\\displaystyle - x y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y z\\\displaystyle 0\\\displaystyle - x y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle z^{2}\\\displaystyle 2 y z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 y z\\\displaystyle y^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle z^{2}\\\displaystyle 0\\\displaystyle 2 x z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y z\\\displaystyle x z\\\displaystyle x y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle 2 x y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 2 x z\\\displaystyle 0\\\displaystyle x^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 2 x y\\\displaystyle x^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y z^{2}\\\displaystyle x z^{2}\\\displaystyle 2 x y z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y^{2} z\\\displaystyle 2 x y z\\\displaystyle x y^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 2 x y z\\\displaystyle x^{2} z\\\displaystyle x^{2} y\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{23}\}$$
• 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 $$e_{0}$$.

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

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 $$e_{0}$$.

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

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 $$e_{1}$$.

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

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 $$e_{1}$$.

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

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 $$e_{2}$$.

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

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 $$e_{2}$$.

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

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 $$e_{3}$$.

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

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

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

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 $$e_{4}$$.

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

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 $$e_{4}$$.

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

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 $$e_{5}$$.

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

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 $$e_{5}$$.

$$\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 2 y \left(- 3 x z + 3 x + z - 1\right)\\\displaystyle 3 x \left(- x z + x + z - 1\right)\\\displaystyle 3 x y \left(1 - x\right)\end{array}\right)$$

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 $$e_{6}$$.

$$\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 3 y z \left(z - 1\right)\\\displaystyle 3 z \left(x z - x - z + 1\right)\\\displaystyle 2 y \left(3 x z - 2 x - 3 z + 2\right)\end{array}\right)$$

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 $$e_{6}$$.

$$\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 3 y z \left(1 - z\right)\\\displaystyle 3 z \left(- x z + x + z - 1\right)\\\displaystyle 2 y \left(- 3 x z + x + 3 z - 1\right)\end{array}\right)$$

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 $$e_{7}$$.

$$\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 3 y z \left(1 - z\right)\\\displaystyle 3 x z \left(1 - z\right)\\\displaystyle 2 x y \left(2 - 3 z\right)\end{array}\right)$$

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 $$e_{7}$$.

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

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 $$e_{8}$$.

$$\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle 2 z \left(3 x y - 3 x - 2 y + 2\right)\\\displaystyle 3 x z \left(x - 1\right)\\\displaystyle 3 x \left(x y - x - y + 1\right)\end{array}\right)$$

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 $$e_{8}$$.

$$\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle 2 z \left(- 3 x y + 3 x + y - 1\right)\\\displaystyle 3 x z \left(1 - x\right)\\\displaystyle 3 x \left(- x y + x + y - 1\right)\end{array}\right)$$

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 $$e_{9}$$.

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

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 $$e_{9}$$.

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

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 $$e_{10}$$.

$$\displaystyle \boldsymbol{\phi}_{20} = \left(\begin{array}{c}\displaystyle 3 y z \left(1 - y\right)\\\displaystyle 2 x z \left(2 - 3 y\right)\\\displaystyle 3 x y \left(1 - y\right)\end{array}\right)$$

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 $$e_{10}$$.

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

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 $$e_{11}$$.

$$\displaystyle \boldsymbol{\phi}_{22} = \left(\begin{array}{c}\displaystyle 2 y z \left(2 - 3 x\right)\\\displaystyle 3 x z \left(1 - x\right)\\\displaystyle 3 x y \left(1 - x\right)\end{array}\right)$$

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 $$e_{11}$$.

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

This DOF is associated with edge 11 of the reference element.
• $$R$$ is the reference hexahedron. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle z^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle z^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y z^{2}\\\displaystyle 0\\\displaystyle - x y z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x z^{2}\\\displaystyle - x y z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y^{2} z\\\displaystyle - x y z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y z\\\displaystyle 0\\\displaystyle - x^{2} y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y z\\\displaystyle - x y^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y z\\\displaystyle - x^{2} z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle z^{3}\\\displaystyle 3 y z^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 3 y^{2} z\\\displaystyle y^{3}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle z^{3}\\\displaystyle 0\\\displaystyle 3 x z^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y z^{2}\\\displaystyle x z^{2}\\\displaystyle 2 x y z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y^{2} z\\\displaystyle 2 x y z\\\displaystyle x y^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y^{3}\\\displaystyle 3 x y^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 2 x y z\\\displaystyle x^{2} z\\\displaystyle x^{2} y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 3 x^{2} z\\\displaystyle 0\\\displaystyle x^{3}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 3 x^{2} y\\\displaystyle x^{3}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y z^{3}\\\displaystyle x z^{3}\\\displaystyle 3 x y z^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y^{3} z\\\displaystyle 3 x y^{2} z\\\displaystyle x y^{3}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 3 x^{2} y z\\\displaystyle x^{3} z\\\displaystyle x^{3} y\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{47}\}$$
• 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 $$e_{0}$$.

$$\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 30 x^{2} y z - 30 x^{2} y - 30 x^{2} z + 30 x^{2} - 36 x y z + 36 x y + 36 x z - 36 x - 3 y^{2} z + 3 y^{2} - 3 y z^{2} + 15 y z - 12 y + 3 z^{2} - 12 z + 9\\\displaystyle 5 x \left(2 x^{2} z - 2 x^{2} - 3 x z + 3 x + z - 1\right)\\\displaystyle 5 x \left(2 x^{2} y - 2 x^{2} - 3 x y + 3 x + y - 1\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 30 x^{2} y z - 30 x^{2} y - 30 x^{2} z + 30 x^{2} - 24 x y z + 24 x y + 24 x z - 24 x - 3 y^{2} z + 3 y^{2} - 3 y z^{2} + 9 y z - 6 y + 3 z^{2} - 6 z + 3\\\displaystyle 5 x \left(2 x^{2} z - 2 x^{2} - 3 x z + 3 x + z - 1\right)\\\displaystyle 5 x \left(2 x^{2} y - 2 x^{2} - 3 x y + 3 x + y - 1\right)\end{array}\right)$$

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

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

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 $$e_{1}$$.

$$\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 5 y \left(2 y^{2} z - 2 y^{2} - 3 y z + 3 y + z - 1\right)\\\displaystyle - 3 x^{2} z + 3 x^{2} + 30 x y^{2} z - 30 x y^{2} - 36 x y z + 36 x y - 3 x z^{2} + 15 x z - 12 x - 30 y^{2} z + 30 y^{2} + 36 y z - 36 y + 3 z^{2} - 12 z + 9\\\displaystyle 5 y \left(2 x y^{2} - 3 x y + x - 2 y^{2} + 3 y - 1\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 5 y \left(2 y^{2} z - 2 y^{2} - 3 y z + 3 y + z - 1\right)\\\displaystyle - 3 x^{2} z + 3 x^{2} + 30 x y^{2} z - 30 x y^{2} - 24 x y z + 24 x y - 3 x z^{2} + 9 x z - 6 x - 30 y^{2} z + 30 y^{2} + 24 y z - 24 y + 3 z^{2} - 6 z + 3\\\displaystyle 5 y \left(2 x y^{2} - 3 x y + x - 2 y^{2} + 3 y - 1\right)\end{array}\right)$$

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

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

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

$$\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 5 z \left(2 y z^{2} - 3 y z + y - 2 z^{2} + 3 z - 1\right)\\\displaystyle 5 z \left(2 x z^{2} - 3 x z + x - 2 z^{2} + 3 z - 1\right)\\\displaystyle - 3 x^{2} y + 3 x^{2} - 3 x y^{2} + 30 x y z^{2} - 36 x y z + 15 x y - 30 x z^{2} + 36 x z - 12 x + 3 y^{2} - 30 y z^{2} + 36 y z - 12 y + 30 z^{2} - 36 z + 9\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 5 z \left(2 y z^{2} - 3 y z + y - 2 z^{2} + 3 z - 1\right)\\\displaystyle 5 z \left(2 x z^{2} - 3 x z + x - 2 z^{2} + 3 z - 1\right)\\\displaystyle - 3 x^{2} y + 3 x^{2} - 3 x y^{2} + 30 x y z^{2} - 24 x y z + 9 x y - 30 x z^{2} + 24 x z - 6 x + 3 y^{2} - 30 y z^{2} + 24 y z - 6 y + 30 z^{2} - 24 z + 3\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle \frac{5 z \left(- 2 y z^{2} + 3 y z - y + 2 z^{2} - 3 z + 1\right)}{2}\\\displaystyle \frac{5 z \left(- 2 x z^{2} + 3 x z - x + 2 z^{2} - 3 z + 1\right)}{2}\\\displaystyle - 3 x^{2} y + 3 x^{2} - 3 x y^{2} - 15 x y z^{2} + 15 x y z + \frac{9 x y}{2} + 15 x z^{2} - 15 x z - \frac{3 x}{2} + 3 y^{2} + 15 y z^{2} - 15 y z - \frac{3 y}{2} - 15 z^{2} + 15 z - \frac{3}{2}\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 5 y \left(- 2 y^{2} z + 2 y^{2} + 3 y z - 3 y - z + 1\right)\\\displaystyle 3 x \left(- x z + x - 10 y^{2} z + 10 y^{2} + 12 y z - 12 y + z^{2} - 3 z + 2\right)\\\displaystyle 5 x y \left(- 2 y^{2} + 3 y - 1\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 5 y \left(- 2 y^{2} z + 2 y^{2} + 3 y z - 3 y - z + 1\right)\\\displaystyle 3 x \left(- x z + x - 10 y^{2} z + 10 y^{2} + 8 y z - 8 y + z^{2} - z\right)\\\displaystyle 5 x y \left(- 2 y^{2} + 3 y - 1\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle \frac{5 y \left(2 y^{2} z - 2 y^{2} - 3 y z + 3 y + z - 1\right)}{2}\\\displaystyle \frac{3 x \left(- 2 x z + 2 x + 10 y^{2} z - 10 y^{2} - 10 y z + 10 y + 2 z^{2} + z - 3\right)}{2}\\\displaystyle \frac{5 x y \left(2 y^{2} - 3 y + 1\right)}{2}\end{array}\right)$$

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{12}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{4}}\boldsymbol{v}\cdot(2 s_{0}^{2} - 3 s_{0} + 1)\hat{\boldsymbol{t}}_{4}$$
where $$e_{4}$$ is the 4th edge;
$$\hat{\boldsymbol{t}}_{4}$$ is the tangent to edge 4;
and $$s_{0}$$ is a parametrisation of $$e_{4}$$.

$$\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 5 z \left(- 2 y z^{2} + 3 y z - y + 2 z^{2} - 3 z + 1\right)\\\displaystyle 5 x z \left(- 2 z^{2} + 3 z - 1\right)\\\displaystyle 3 x \left(- x y + x + y^{2} - 10 y z^{2} + 12 y z - 3 y + 10 z^{2} - 12 z + 2\right)\end{array}\right)$$

This DOF is associated with edge 4 of the reference element.
$$\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{4}}\boldsymbol{v}\cdot(s_{0} \cdot \left(2 s_{0} - 1\right))\hat{\boldsymbol{t}}_{4}$$
where $$e_{4}$$ is the 4th edge;
$$\hat{\boldsymbol{t}}_{4}$$ is the tangent to edge 4;
and $$s_{0}$$ is a parametrisation of $$e_{4}$$.

$$\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 5 z \left(- 2 y z^{2} + 3 y z - y + 2 z^{2} - 3 z + 1\right)\\\displaystyle 5 x z \left(- 2 z^{2} + 3 z - 1\right)\\\displaystyle 3 x \left(- x y + x + y^{2} - 10 y z^{2} + 8 y z - y + 10 z^{2} - 8 z\right)\end{array}\right)$$

This DOF is associated with edge 4 of the reference element.
$$\displaystyle l_{14}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{4}}\boldsymbol{v}\cdot(4 s_{0} \cdot \left(1 - s_{0}\right))\hat{\boldsymbol{t}}_{4}$$
where $$e_{4}$$ is the 4th edge;
$$\hat{\boldsymbol{t}}_{4}$$ is the tangent to edge 4;
and $$s_{0}$$ is a parametrisation of $$e_{4}$$.

$$\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle \frac{5 z \left(2 y z^{2} - 3 y z + y - 2 z^{2} + 3 z - 1\right)}{2}\\\displaystyle \frac{5 x z \left(2 z^{2} - 3 z + 1\right)}{2}\\\displaystyle \frac{3 x \left(- 2 x y + 2 x + 2 y^{2} + 10 y z^{2} - 10 y z + y - 10 z^{2} + 10 z - 3\right)}{2}\end{array}\right)$$

This DOF is associated with edge 4 of the reference element.
$$\displaystyle l_{15}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{5}}\boldsymbol{v}\cdot(2 s_{0}^{2} - 3 s_{0} + 1)\hat{\boldsymbol{t}}_{5}$$
where $$e_{5}$$ is the 5th edge;
$$\hat{\boldsymbol{t}}_{5}$$ is the tangent to edge 5;
and $$s_{0}$$ is a parametrisation of $$e_{5}$$.

$$\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 3 y \left(- 10 x^{2} z + 10 x^{2} + 12 x z - 12 x - y z + y + z^{2} - 3 z + 2\right)\\\displaystyle 5 x \left(- 2 x^{2} z + 2 x^{2} + 3 x z - 3 x - z + 1\right)\\\displaystyle 5 x y \left(- 2 x^{2} + 3 x - 1\right)\end{array}\right)$$

This DOF is associated with edge 5 of the reference element.
$$\displaystyle l_{16}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{5}}\boldsymbol{v}\cdot(s_{0} \cdot \left(2 s_{0} - 1\right))\hat{\boldsymbol{t}}_{5}$$
where $$e_{5}$$ is the 5th edge;
$$\hat{\boldsymbol{t}}_{5}$$ is the tangent to edge 5;
and $$s_{0}$$ is a parametrisation of $$e_{5}$$.

$$\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle 3 y \left(- 10 x^{2} z + 10 x^{2} + 8 x z - 8 x - y z + y + z^{2} - z\right)\\\displaystyle 5 x \left(- 2 x^{2} z + 2 x^{2} + 3 x z - 3 x - z + 1\right)\\\displaystyle 5 x y \left(- 2 x^{2} + 3 x - 1\right)\end{array}\right)$$

This DOF is associated with edge 5 of the reference element.
$$\displaystyle l_{17}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{5}}\boldsymbol{v}\cdot(4 s_{0} \cdot \left(1 - s_{0}\right))\hat{\boldsymbol{t}}_{5}$$
where $$e_{5}$$ is the 5th edge;
$$\hat{\boldsymbol{t}}_{5}$$ is the tangent to edge 5;
and $$s_{0}$$ is a parametrisation of $$e_{5}$$.

$$\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle \frac{3 y \left(10 x^{2} z - 10 x^{2} - 10 x z + 10 x - 2 y z + 2 y + 2 z^{2} + z - 3\right)}{2}\\\displaystyle \frac{5 x \left(2 x^{2} z - 2 x^{2} - 3 x z + 3 x + z - 1\right)}{2}\\\displaystyle \frac{5 x y \left(2 x^{2} - 3 x + 1\right)}{2}\end{array}\right)$$

This DOF is associated with edge 5 of the reference element.
$$\displaystyle l_{18}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{6}}\boldsymbol{v}\cdot(2 s_{0}^{2} - 3 s_{0} + 1)\hat{\boldsymbol{t}}_{6}$$
where $$e_{6}$$ is the 6th edge;
$$\hat{\boldsymbol{t}}_{6}$$ is the tangent to edge 6;
and $$s_{0}$$ is a parametrisation of $$e_{6}$$.

$$\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle 5 y z \left(- 2 z^{2} + 3 z - 1\right)\\\displaystyle 5 z \left(- 2 x z^{2} + 3 x z - x + 2 z^{2} - 3 z + 1\right)\\\displaystyle 3 y \left(x^{2} - x y - 10 x z^{2} + 12 x z - 3 x + y + 10 z^{2} - 12 z + 2\right)\end{array}\right)$$

This DOF is associated with edge 6 of the reference element.
$$\displaystyle l_{19}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{6}}\boldsymbol{v}\cdot(s_{0} \cdot \left(2 s_{0} - 1\right))\hat{\boldsymbol{t}}_{6}$$
where $$e_{6}$$ is the 6th edge;
$$\hat{\boldsymbol{t}}_{6}$$ is the tangent to edge 6;
and $$s_{0}$$ is a parametrisation of $$e_{6}$$.

$$\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle 5 y z \left(- 2 z^{2} + 3 z - 1\right)\\\displaystyle 5 z \left(- 2 x z^{2} + 3 x z - x + 2 z^{2} - 3 z + 1\right)\\\displaystyle 3 y \left(x^{2} - x y - 10 x z^{2} + 8 x z - x + y + 10 z^{2} - 8 z\right)\end{array}\right)$$

This DOF is associated with edge 6 of the reference element.
$$\displaystyle l_{20}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{6}}\boldsymbol{v}\cdot(4 s_{0} \cdot \left(1 - s_{0}\right))\hat{\boldsymbol{t}}_{6}$$
where $$e_{6}$$ is the 6th edge;
$$\hat{\boldsymbol{t}}_{6}$$ is the tangent to edge 6;
and $$s_{0}$$ is a parametrisation of $$e_{6}$$.

$$\displaystyle \boldsymbol{\phi}_{20} = \left(\begin{array}{c}\displaystyle \frac{5 y z \left(2 z^{2} - 3 z + 1\right)}{2}\\\displaystyle \frac{5 z \left(2 x z^{2} - 3 x z + x - 2 z^{2} + 3 z - 1\right)}{2}\\\displaystyle \frac{3 y \left(2 x^{2} - 2 x y + 10 x z^{2} - 10 x z + x + 2 y - 10 z^{2} + 10 z - 3\right)}{2}\end{array}\right)$$

This DOF is associated with edge 6 of the reference element.
$$\displaystyle l_{21}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{7}}\boldsymbol{v}\cdot(2 s_{0}^{2} - 3 s_{0} + 1)\hat{\boldsymbol{t}}_{7}$$
where $$e_{7}$$ is the 7th edge;
$$\hat{\boldsymbol{t}}_{7}$$ is the tangent to edge 7;
and $$s_{0}$$ is a parametrisation of $$e_{7}$$.

$$\displaystyle \boldsymbol{\phi}_{21} = \left(\begin{array}{c}\displaystyle 5 y z \left(2 z^{2} - 3 z + 1\right)\\\displaystyle 5 x z \left(2 z^{2} - 3 z + 1\right)\\\displaystyle 3 x y \left(x + y + 10 z^{2} - 12 z + 1\right)\end{array}\right)$$

This DOF is associated with edge 7 of the reference element.
$$\displaystyle l_{22}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{7}}\boldsymbol{v}\cdot(s_{0} \cdot \left(2 s_{0} - 1\right))\hat{\boldsymbol{t}}_{7}$$
where $$e_{7}$$ is the 7th edge;
$$\hat{\boldsymbol{t}}_{7}$$ is the tangent to edge 7;
and $$s_{0}$$ is a parametrisation of $$e_{7}$$.

$$\displaystyle \boldsymbol{\phi}_{22} = \left(\begin{array}{c}\displaystyle 5 y z \left(2 z^{2} - 3 z + 1\right)\\\displaystyle 5 x z \left(2 z^{2} - 3 z + 1\right)\\\displaystyle 3 x y \left(x + y + 10 z^{2} - 8 z - 1\right)\end{array}\right)$$

This DOF is associated with edge 7 of the reference element.
$$\displaystyle l_{23}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{7}}\boldsymbol{v}\cdot(4 s_{0} \cdot \left(1 - s_{0}\right))\hat{\boldsymbol{t}}_{7}$$
where $$e_{7}$$ is the 7th edge;
$$\hat{\boldsymbol{t}}_{7}$$ is the tangent to edge 7;
and $$s_{0}$$ is a parametrisation of $$e_{7}$$.

$$\displaystyle \boldsymbol{\phi}_{23} = \left(\begin{array}{c}\displaystyle \frac{5 y z \left(- 2 z^{2} + 3 z - 1\right)}{2}\\\displaystyle \frac{5 x z \left(- 2 z^{2} + 3 z - 1\right)}{2}\\\displaystyle \frac{3 x y \left(2 x + 2 y - 10 z^{2} + 10 z - 5\right)}{2}\end{array}\right)$$

This DOF is associated with edge 7 of the reference element.
$$\displaystyle l_{24}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{8}}\boldsymbol{v}\cdot(2 s_{0}^{2} - 3 s_{0} + 1)\hat{\boldsymbol{t}}_{8}$$
where $$e_{8}$$ is the 8th edge;
$$\hat{\boldsymbol{t}}_{8}$$ is the tangent to edge 8;
and $$s_{0}$$ is a parametrisation of $$e_{8}$$.

$$\displaystyle \boldsymbol{\phi}_{24} = \left(\begin{array}{c}\displaystyle 3 z \left(- 10 x^{2} y + 10 x^{2} + 12 x y - 12 x + y^{2} - y z - 3 y + z + 2\right)\\\displaystyle 5 x z \left(- 2 x^{2} + 3 x - 1\right)\\\displaystyle 5 x \left(- 2 x^{2} y + 2 x^{2} + 3 x y - 3 x - y + 1\right)\end{array}\right)$$

This DOF is associated with edge 8 of the reference element.
$$\displaystyle l_{25}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{8}}\boldsymbol{v}\cdot(s_{0} \cdot \left(2 s_{0} - 1\right))\hat{\boldsymbol{t}}_{8}$$
where $$e_{8}$$ is the 8th edge;
$$\hat{\boldsymbol{t}}_{8}$$ is the tangent to edge 8;
and $$s_{0}$$ is a parametrisation of $$e_{8}$$.

$$\displaystyle \boldsymbol{\phi}_{25} = \left(\begin{array}{c}\displaystyle 3 z \left(- 10 x^{2} y + 10 x^{2} + 8 x y - 8 x + y^{2} - y z - y + z\right)\\\displaystyle 5 x z \left(- 2 x^{2} + 3 x - 1\right)\\\displaystyle 5 x \left(- 2 x^{2} y + 2 x^{2} + 3 x y - 3 x - y + 1\right)\end{array}\right)$$

This DOF is associated with edge 8 of the reference element.
$$\displaystyle l_{26}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{8}}\boldsymbol{v}\cdot(4 s_{0} \cdot \left(1 - s_{0}\right))\hat{\boldsymbol{t}}_{8}$$
where $$e_{8}$$ is the 8th edge;
$$\hat{\boldsymbol{t}}_{8}$$ is the tangent to edge 8;
and $$s_{0}$$ is a parametrisation of $$e_{8}$$.

$$\displaystyle \boldsymbol{\phi}_{26} = \left(\begin{array}{c}\displaystyle \frac{3 z \left(10 x^{2} y - 10 x^{2} - 10 x y + 10 x + 2 y^{2} - 2 y z + y + 2 z - 3\right)}{2}\\\displaystyle \frac{5 x z \left(2 x^{2} - 3 x + 1\right)}{2}\\\displaystyle \frac{5 x \left(2 x^{2} y - 2 x^{2} - 3 x y + 3 x + y - 1\right)}{2}\end{array}\right)$$

This DOF is associated with edge 8 of the reference element.
$$\displaystyle l_{27}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{9}}\boldsymbol{v}\cdot(2 s_{0}^{2} - 3 s_{0} + 1)\hat{\boldsymbol{t}}_{9}$$
where $$e_{9}$$ is the 9th edge;
$$\hat{\boldsymbol{t}}_{9}$$ is the tangent to edge 9;
and $$s_{0}$$ is a parametrisation of $$e_{9}$$.

$$\displaystyle \boldsymbol{\phi}_{27} = \left(\begin{array}{c}\displaystyle 5 y z \left(- 2 y^{2} + 3 y - 1\right)\\\displaystyle 3 z \left(x^{2} - 10 x y^{2} + 12 x y - x z - 3 x + 10 y^{2} - 12 y + z + 2\right)\\\displaystyle 5 y \left(- 2 x y^{2} + 3 x y - x + 2 y^{2} - 3 y + 1\right)\end{array}\right)$$

This DOF is associated with edge 9 of the reference element.
$$\displaystyle l_{28}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{9}}\boldsymbol{v}\cdot(s_{0} \cdot \left(2 s_{0} - 1\right))\hat{\boldsymbol{t}}_{9}$$
where $$e_{9}$$ is the 9th edge;
$$\hat{\boldsymbol{t}}_{9}$$ is the tangent to edge 9;
and $$s_{0}$$ is a parametrisation of $$e_{9}$$.

$$\displaystyle \boldsymbol{\phi}_{28} = \left(\begin{array}{c}\displaystyle 5 y z \left(- 2 y^{2} + 3 y - 1\right)\\\displaystyle 3 z \left(x^{2} - 10 x y^{2} + 8 x y - x z - x + 10 y^{2} - 8 y + z\right)\\\displaystyle 5 y \left(- 2 x y^{2} + 3 x y - x + 2 y^{2} - 3 y + 1\right)\end{array}\right)$$

This DOF is associated with edge 9 of the reference element.
$$\displaystyle l_{29}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{9}}\boldsymbol{v}\cdot(4 s_{0} \cdot \left(1 - s_{0}\right))\hat{\boldsymbol{t}}_{9}$$
where $$e_{9}$$ is the 9th edge;
$$\hat{\boldsymbol{t}}_{9}$$ is the tangent to edge 9;
and $$s_{0}$$ is a parametrisation of $$e_{9}$$.

$$\displaystyle \boldsymbol{\phi}_{29} = \left(\begin{array}{c}\displaystyle \frac{5 y z \left(2 y^{2} - 3 y + 1\right)}{2}\\\displaystyle \frac{3 z \left(2 x^{2} + 10 x y^{2} - 10 x y - 2 x z + x - 10 y^{2} + 10 y + 2 z - 3\right)}{2}\\\displaystyle \frac{5 y \left(2 x y^{2} - 3 x y + x - 2 y^{2} + 3 y - 1\right)}{2}\end{array}\right)$$

This DOF is associated with edge 9 of the reference element.
$$\displaystyle l_{30}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{10}}\boldsymbol{v}\cdot(2 s_{0}^{2} - 3 s_{0} + 1)\hat{\boldsymbol{t}}_{10}$$
where $$e_{10}$$ is the 10th edge;
$$\hat{\boldsymbol{t}}_{10}$$ is the tangent to edge 10;
and $$s_{0}$$ is a parametrisation of $$e_{10}$$.

$$\displaystyle \boldsymbol{\phi}_{30} = \left(\begin{array}{c}\displaystyle 5 y z \left(2 y^{2} - 3 y + 1\right)\\\displaystyle 3 x z \left(x + 10 y^{2} - 12 y + z + 1\right)\\\displaystyle 5 x y \left(2 y^{2} - 3 y + 1\right)\end{array}\right)$$

This DOF is associated with edge 10 of the reference element.
$$\displaystyle l_{31}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{10}}\boldsymbol{v}\cdot(s_{0} \cdot \left(2 s_{0} - 1\right))\hat{\boldsymbol{t}}_{10}$$
where $$e_{10}$$ is the 10th edge;
$$\hat{\boldsymbol{t}}_{10}$$ is the tangent to edge 10;
and $$s_{0}$$ is a parametrisation of $$e_{10}$$.

$$\displaystyle \boldsymbol{\phi}_{31} = \left(\begin{array}{c}\displaystyle 5 y z \left(2 y^{2} - 3 y + 1\right)\\\displaystyle 3 x z \left(x + 10 y^{2} - 8 y + z - 1\right)\\\displaystyle 5 x y \left(2 y^{2} - 3 y + 1\right)\end{array}\right)$$

This DOF is associated with edge 10 of the reference element.
$$\displaystyle l_{32}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{10}}\boldsymbol{v}\cdot(4 s_{0} \cdot \left(1 - s_{0}\right))\hat{\boldsymbol{t}}_{10}$$
where $$e_{10}$$ is the 10th edge;
$$\hat{\boldsymbol{t}}_{10}$$ is the tangent to edge 10;
and $$s_{0}$$ is a parametrisation of $$e_{10}$$.

$$\displaystyle \boldsymbol{\phi}_{32} = \left(\begin{array}{c}\displaystyle \frac{5 y z \left(- 2 y^{2} + 3 y - 1\right)}{2}\\\displaystyle \frac{3 x z \left(2 x - 10 y^{2} + 10 y + 2 z - 5\right)}{2}\\\displaystyle \frac{5 x y \left(- 2 y^{2} + 3 y - 1\right)}{2}\end{array}\right)$$

This DOF is associated with edge 10 of the reference element.
$$\displaystyle l_{33}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{11}}\boldsymbol{v}\cdot(2 s_{0}^{2} - 3 s_{0} + 1)\hat{\boldsymbol{t}}_{11}$$
where $$e_{11}$$ is the 11th edge;
$$\hat{\boldsymbol{t}}_{11}$$ is the tangent to edge 11;
and $$s_{0}$$ is a parametrisation of $$e_{11}$$.

$$\displaystyle \boldsymbol{\phi}_{33} = \left(\begin{array}{c}\displaystyle 3 y z \left(10 x^{2} - 12 x + y + z + 1\right)\\\displaystyle 5 x z \left(2 x^{2} - 3 x + 1\right)\\\displaystyle 5 x y \left(2 x^{2} - 3 x + 1\right)\end{array}\right)$$

This DOF is associated with edge 11 of the reference element.
$$\displaystyle l_{34}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{11}}\boldsymbol{v}\cdot(s_{0} \cdot \left(2 s_{0} - 1\right))\hat{\boldsymbol{t}}_{11}$$
where $$e_{11}$$ is the 11th edge;
$$\hat{\boldsymbol{t}}_{11}$$ is the tangent to edge 11;
and $$s_{0}$$ is a parametrisation of $$e_{11}$$.

$$\displaystyle \boldsymbol{\phi}_{34} = \left(\begin{array}{c}\displaystyle 3 y z \left(10 x^{2} - 8 x + y + z - 1\right)\\\displaystyle 5 x z \left(2 x^{2} - 3 x + 1\right)\\\displaystyle 5 x y \left(2 x^{2} - 3 x + 1\right)\end{array}\right)$$

This DOF is associated with edge 11 of the reference element.
$$\displaystyle l_{35}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{11}}\boldsymbol{v}\cdot(4 s_{0} \cdot \left(1 - s_{0}\right))\hat{\boldsymbol{t}}_{11}$$
where $$e_{11}$$ is the 11th edge;
$$\hat{\boldsymbol{t}}_{11}$$ is the tangent to edge 11;
and $$s_{0}$$ is a parametrisation of $$e_{11}$$.

$$\displaystyle \boldsymbol{\phi}_{35} = \left(\begin{array}{c}\displaystyle \frac{3 y z \left(- 10 x^{2} + 10 x + 2 y + 2 z - 5\right)}{2}\\\displaystyle \frac{5 x z \left(- 2 x^{2} + 3 x - 1\right)}{2}\\\displaystyle \frac{5 x y \left(- 2 x^{2} + 3 x - 1\right)}{2}\end{array}\right)$$

This DOF is associated with edge 11 of the reference element.
$$\displaystyle l_{36}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)$$
where $$f_{0}$$ is the 0th face.

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

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

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

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

$$\displaystyle \boldsymbol{\phi}_{38} = \left(\begin{array}{c}\displaystyle 6 z \left(y z - y - z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

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

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

This DOF is associated with face 1 of the reference element.
$$\displaystyle l_{40}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)$$
where $$f_{2}$$ is the 2nd face.

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

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{41}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)$$
where $$f_{2}$$ is the 2nd face.

$$\displaystyle \boldsymbol{\phi}_{41} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 6 y \left(x y - x - y + 1\right)\end{array}\right)$$

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{42}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)$$
where $$f_{3}$$ is the 3th face.

$$\displaystyle \boldsymbol{\phi}_{42} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 6 x z \left(1 - z\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 3 of the reference element.
$$\displaystyle l_{43}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)$$
where $$f_{3}$$ is the 3th face.

$$\displaystyle \boldsymbol{\phi}_{43} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 6 x y \left(1 - y\right)\end{array}\right)$$

This DOF is associated with face 3 of the reference element.
$$\displaystyle l_{44}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)$$
where $$f_{4}$$ is the 4th face.

$$\displaystyle \boldsymbol{\phi}_{44} = \left(\begin{array}{c}\displaystyle 6 y z \left(1 - z\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 4 of the reference element.
$$\displaystyle l_{45}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)$$
where $$f_{4}$$ is the 4th face.

$$\displaystyle \boldsymbol{\phi}_{45} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 6 x y \left(1 - x\right)\end{array}\right)$$

This DOF is associated with face 4 of the reference element.
$$\displaystyle l_{46}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)$$
where $$f_{5}$$ is the 5th face.

$$\displaystyle \boldsymbol{\phi}_{46} = \left(\begin{array}{c}\displaystyle 6 y z \left(1 - y\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 5 of the reference element.
$$\displaystyle l_{47}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)$$
where $$f_{5}$$ is the 5th face.

$$\displaystyle \boldsymbol{\phi}_{47} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 6 x z \left(1 - x\right)\\\displaystyle 0\end{array}\right)$$

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

## DefElement stats

 Element added 31 December 2020 Element last updated 10 February 2022