an encyclopedia of finite element definitions

# Serendipity H(div)

 Alternative names Brezzi–Douglas–Marini cubical H(div) (quadrilateral), Arnold–Awanou H(div) (hexahedron), $$\mathcal{S}_k\Lambda^{d-1}(\square_d)$$ Abbreviated names BDMcf (quadrilateral), AAf (hexahedron) Orders $$1\leqslant k$$ Reference elements quadrilateral, hexahedron Polynomial set $$\mathcal{P}_{k}^d \oplus \mathcal{Z}^{(2)}_{k}$$ (quadrilateral) $$\mathcal{P}_{k}^d \oplus \mathcal{Z}^{(3)}_{k}$$ (hexahedron) ↓ Show polynomial set definitions ↓ DOFs On each facet: normal integral moments with an order $$k$$ dPc space On the interior of the reference element: integral moments with an order $$k-2$$ vector dPc space Number of DOFs quadrilateral: $$k^2+3k+4$$ (A014206)hexahedron: $$(k+1)(k^2+5k+12)/2$$ Categories Vector-valued elements, H(div) conforming elements

## Implementations

 Symfem string "Sdiv"↓ Show Symfem examples ↓ UFL string "BDMCF" (quadrilateral)"AAF" (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: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 2 x y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 2 x y\\\displaystyle y^{2}\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{7}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(1 - t_{0})\hat{\boldsymbol{n}}_{0}$$

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

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(t_{0})\hat{\boldsymbol{n}}_{0}$$

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

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(1 - t_{0})\hat{\boldsymbol{n}}_{1}$$

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

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(t_{0})\hat{\boldsymbol{n}}_{1}$$

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

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(1 - t_{0})\hat{\boldsymbol{n}}_{2}$$

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

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(t_{0})\hat{\boldsymbol{n}}_{2}$$

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

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(1 - t_{0})\hat{\boldsymbol{n}}_{3}$$

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

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(t_{0})\hat{\boldsymbol{n}}_{3}$$

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

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: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{3}\\\displaystyle 3 x^{2} y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 3 x y^{2}\\\displaystyle y^{3}\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{13}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(2 t_{0}^{2} - 3 t_{0} + 1)\hat{\boldsymbol{n}}_{0}$$

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

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(4 t_{0} \left(1 - t_{0}\right))\hat{\boldsymbol{n}}_{0}$$

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

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(t_{0} \left(2 t_{0} - 1\right))\hat{\boldsymbol{n}}_{0}$$

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

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(2 t_{0}^{2} - 3 t_{0} + 1)\hat{\boldsymbol{n}}_{1}$$

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

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(4 t_{0} \left(1 - t_{0}\right))\hat{\boldsymbol{n}}_{1}$$

$$\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle - 3 x^{2} - 15 x y^{2} + 15 x y + \frac{3 x}{2} + 15 y^{2} - 15 y + \frac{3}{2}\\\displaystyle \frac{5 y \left(- 2 y^{2} + 3 y - 1\right)}{2}\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(t_{0} \left(2 t_{0} - 1\right))\hat{\boldsymbol{n}}_{1}$$

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

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(2 t_{0}^{2} - 3 t_{0} + 1)\hat{\boldsymbol{n}}_{2}$$

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

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(4 t_{0} \left(1 - t_{0}\right))\hat{\boldsymbol{n}}_{2}$$

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

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(t_{0} \left(2 t_{0} - 1\right))\hat{\boldsymbol{n}}_{2}$$

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

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{9}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(2 t_{0}^{2} - 3 t_{0} + 1)\hat{\boldsymbol{n}}_{3}$$

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

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{10}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(4 t_{0} \left(1 - t_{0}\right))\hat{\boldsymbol{n}}_{3}$$

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

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{11}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(t_{0} \left(2 t_{0} - 1\right))\hat{\boldsymbol{n}}_{3}$$

$$\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 5 x \left(2 x^{2} - 3 x + 1\right)\\\displaystyle 3 y \left(10 x^{2} - 8 x + y\right)\end{array}\right)$$

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

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

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

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

This DOF is associated with face 0 of the reference element.
• $$R$$ is the reference hexahedron. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle - 3 x z\\\displaystyle y z\\\displaystyle z^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle - x z\\\displaystyle 3 y z\\\displaystyle - z^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y\\\displaystyle y^{2}\\\displaystyle - 3 y z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle - 3 x y\\\displaystyle y^{2}\\\displaystyle y z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle - x^{2}\\\displaystyle 3 x y\\\displaystyle - x z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle x y\\\displaystyle - 3 x z\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{17}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(- t_{0} - t_{1} + 1)\hat{\boldsymbol{n}}_{0}$$

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

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(t_{0})\hat{\boldsymbol{n}}_{0}$$

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

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(t_{1})\hat{\boldsymbol{n}}_{0}$$

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

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(- t_{0} - t_{1} + 1)\hat{\boldsymbol{n}}_{1}$$

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

This DOF is associated with face 1 of the reference element.
$$\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(t_{0})\hat{\boldsymbol{n}}_{1}$$

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

This DOF is associated with face 1 of the reference element.
$$\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(t_{1})\hat{\boldsymbol{n}}_{1}$$

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

This DOF is associated with face 1 of the reference element.
$$\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(- t_{0} - t_{1} + 1)\hat{\boldsymbol{n}}_{2}$$

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

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(t_{0})\hat{\boldsymbol{n}}_{2}$$

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

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(t_{1})\hat{\boldsymbol{n}}_{2}$$

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

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{9}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(- t_{0} - t_{1} + 1)\hat{\boldsymbol{n}}_{3}$$

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

This DOF is associated with face 3 of the reference element.
$$\displaystyle l_{10}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(t_{0})\hat{\boldsymbol{n}}_{3}$$

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

This DOF is associated with face 3 of the reference element.
$$\displaystyle l_{11}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(t_{1})\hat{\boldsymbol{n}}_{3}$$

$$\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle x \left(- 6 y + 6 z + 1\right)\\\displaystyle 3 y \left(y - 1\right)\\\displaystyle 3 z \left(1 - z\right)\end{array}\right)$$

This DOF is associated with face 3 of the reference element.
$$\displaystyle l_{12}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot(- t_{0} - t_{1} + 1)\hat{\boldsymbol{n}}_{4}$$

$$\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 3 x \left(1 - x\right)\\\displaystyle y \left(6 x + 6 z - 7\right)\\\displaystyle 3 z \left(1 - z\right)\end{array}\right)$$

This DOF is associated with face 4 of the reference element.
$$\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot(t_{0})\hat{\boldsymbol{n}}_{4}$$

$$\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 3 x \left(x - 1\right)\\\displaystyle y \left(- 6 x + 6 z - 1\right)\\\displaystyle 3 z \left(1 - z\right)\end{array}\right)$$

This DOF is associated with face 4 of the reference element.
$$\displaystyle l_{14}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot(t_{1})\hat{\boldsymbol{n}}_{4}$$

$$\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 3 x \left(1 - x\right)\\\displaystyle y \left(6 x - 6 z - 1\right)\\\displaystyle 3 z \left(z - 1\right)\end{array}\right)$$

This DOF is associated with face 4 of the reference element.
$$\displaystyle l_{15}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot(- t_{0} - t_{1} + 1)\hat{\boldsymbol{n}}_{5}$$

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

This DOF is associated with face 5 of the reference element.
$$\displaystyle l_{16}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot(t_{0})\hat{\boldsymbol{n}}_{5}$$

$$\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle 3 x \left(1 - x\right)\\\displaystyle 3 y \left(y - 1\right)\\\displaystyle z \left(6 x - 6 y + 1\right)\end{array}\right)$$

This DOF is associated with face 5 of the reference element.
$$\displaystyle l_{17}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot(t_{1})\hat{\boldsymbol{n}}_{5}$$

$$\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle 3 x \left(x - 1\right)\\\displaystyle 3 y \left(1 - y\right)\\\displaystyle z \left(- 6 x + 6 y + 1\right)\end{array}\right)$$

This DOF is associated with face 5 of the reference element.
• $$R$$ is the reference hexahedron. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle z^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle z^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle - 4 x z^{2}\\\displaystyle y z^{2}\\\displaystyle z^{3}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle - x z^{2}\\\displaystyle 4 y z^{2}\\\displaystyle - z^{3}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y^{2}\\\displaystyle y^{3}\\\displaystyle - 4 y^{2} z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle - 4 x y z\\\displaystyle y^{2} z\\\displaystyle y z^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle - x^{2} z\\\displaystyle 4 x y z\\\displaystyle - x z^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2} y\\\displaystyle x y^{2}\\\displaystyle - 4 x y z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle - 4 x y^{2}\\\displaystyle y^{3}\\\displaystyle y^{2} z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle - x^{3}\\\displaystyle 4 x^{2} y\\\displaystyle - x^{2} z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{3}\\\displaystyle x^{2} y\\\displaystyle - 4 x^{2} z\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{38}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(2 t_{0}^{2} + 4 t_{0} t_{1} - 3 t_{0} + 2 t_{1}^{2} - 3 t_{1} + 1)\hat{\boldsymbol{n}}_{0}$$

$$\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle \frac{x \left(20 x^{2} + 18 x y - 39 x - 18 y + 19\right)}{2}\\\displaystyle \frac{y \left(18 x y - 18 x + 20 y^{2} - 39 y + 19\right)}{2}\\\displaystyle - 30 x^{2} z + 30 x^{2} - 36 x y z + 36 x y + 54 x z - 54 x - 30 y^{2} z + 30 y^{2} + 54 y z - 54 y + 3 z^{2} - 29 z + 26\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(4 t_{0} \left(- t_{0} - t_{1} + 1\right))\hat{\boldsymbol{n}}_{0}$$

$$\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle \frac{5 x \left(- 2 x^{2} + 3 x - 1\right)}{2}\\\displaystyle 5 y \left(2 y^{2} - 3 y + 1\right)\\\displaystyle 15 x^{2} z - 15 x^{2} - 15 x z + 15 x - 30 y^{2} z + 30 y^{2} + 36 y z - 36 y + 3 z^{2} - \frac{19 z}{2} + \frac{13}{2}\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(t_{0} \left(2 t_{0} - 1\right))\hat{\boldsymbol{n}}_{0}$$

$$\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle \frac{x \left(20 x^{2} - 18 x y - 21 x + 18 y + 1\right)}{2}\\\displaystyle \frac{y \left(- 18 x y + 18 x + 20 y^{2} - 21 y + 1\right)}{2}\\\displaystyle - 30 x^{2} z + 30 x^{2} + 36 x y z - 36 x y + 6 x z - 6 x - 30 y^{2} z + 30 y^{2} + 18 y z - 18 y + 3 z^{2} - 5 z + 2\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(4 t_{1} \left(- t_{0} - t_{1} + 1\right))\hat{\boldsymbol{n}}_{0}$$

$$\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 5 x \left(2 x^{2} - 3 x + 1\right)\\\displaystyle \frac{5 y \left(- 2 y^{2} + 3 y - 1\right)}{2}\\\displaystyle - 30 x^{2} z + 30 x^{2} + 36 x z - 36 x + 15 y^{2} z - 15 y^{2} - 15 y z + 15 y + 3 z^{2} - \frac{19 z}{2} + \frac{13}{2}\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(4 t_{0} t_{1})\hat{\boldsymbol{n}}_{0}$$

$$\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle \frac{5 x \left(- 2 x^{2} + 3 x - 1\right)}{2}\\\displaystyle \frac{5 y \left(- 2 y^{2} + 3 y - 1\right)}{2}\\\displaystyle 15 x^{2} z - 15 x^{2} - 15 x z + 15 x + 15 y^{2} z - 15 y^{2} - 15 y z + 15 y + 3 z^{2} + z - 4\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(t_{1} \left(2 t_{1} - 1\right))\hat{\boldsymbol{n}}_{0}$$

$$\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle \frac{x \left(20 x^{2} - 18 x y - 21 x + 18 y + 1\right)}{2}\\\displaystyle \frac{y \left(- 18 x y + 18 x + 20 y^{2} - 21 y + 1\right)}{2}\\\displaystyle - 30 x^{2} z + 30 x^{2} + 36 x y z - 36 x y + 18 x z - 18 x - 30 y^{2} z + 30 y^{2} + 6 y z - 6 y + 3 z^{2} - 5 z + 2\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(2 t_{0}^{2} + 4 t_{0} t_{1} - 3 t_{0} + 2 t_{1}^{2} - 3 t_{1} + 1)\hat{\boldsymbol{n}}_{1}$$

$$\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle \frac{x \left(- 20 x^{2} - 18 x z + 39 x + 18 z - 19\right)}{2}\\\displaystyle 30 x^{2} y - 30 x^{2} + 36 x y z - 54 x y - 36 x z + 54 x - 3 y^{2} + 30 y z^{2} - 54 y z + 29 y - 30 z^{2} + 54 z - 26\\\displaystyle \frac{z \left(- 18 x z + 18 x - 20 z^{2} + 39 z - 19\right)}{2}\end{array}\right)$$

This DOF is associated with face 1 of the reference element.
$$\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(4 t_{0} \left(- t_{0} - t_{1} + 1\right))\hat{\boldsymbol{n}}_{1}$$

$$\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle \frac{5 x \left(2 x^{2} - 3 x + 1\right)}{2}\\\displaystyle - 15 x^{2} y + 15 x^{2} + 15 x y - 15 x - 3 y^{2} + 30 y z^{2} - 36 y z + \frac{19 y}{2} - 30 z^{2} + 36 z - \frac{13}{2}\\\displaystyle 5 z \left(- 2 z^{2} + 3 z - 1\right)\end{array}\right)$$

This DOF is associated with face 1 of the reference element.
$$\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(t_{0} \left(2 t_{0} - 1\right))\hat{\boldsymbol{n}}_{1}$$

$$\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle \frac{x \left(- 20 x^{2} + 18 x z + 21 x - 18 z - 1\right)}{2}\\\displaystyle 30 x^{2} y - 30 x^{2} - 36 x y z - 6 x y + 36 x z + 6 x - 3 y^{2} + 30 y z^{2} - 18 y z + 5 y - 30 z^{2} + 18 z - 2\\\displaystyle \frac{z \left(18 x z - 18 x - 20 z^{2} + 21 z - 1\right)}{2}\end{array}\right)$$

This DOF is associated with face 1 of the reference element.
$$\displaystyle l_{9}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(4 t_{1} \left(- t_{0} - t_{1} + 1\right))\hat{\boldsymbol{n}}_{1}$$

$$\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 5 x \left(- 2 x^{2} + 3 x - 1\right)\\\displaystyle 30 x^{2} y - 30 x^{2} - 36 x y + 36 x - 3 y^{2} - 15 y z^{2} + 15 y z + \frac{19 y}{2} + 15 z^{2} - 15 z - \frac{13}{2}\\\displaystyle \frac{5 z \left(2 z^{2} - 3 z + 1\right)}{2}\end{array}\right)$$

This DOF is associated with face 1 of the reference element.
$$\displaystyle l_{10}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(4 t_{0} t_{1})\hat{\boldsymbol{n}}_{1}$$

$$\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle \frac{5 x \left(2 x^{2} - 3 x + 1\right)}{2}\\\displaystyle - 15 x^{2} y + 15 x^{2} + 15 x y - 15 x - 3 y^{2} - 15 y z^{2} + 15 y z - y + 15 z^{2} - 15 z + 4\\\displaystyle \frac{5 z \left(2 z^{2} - 3 z + 1\right)}{2}\end{array}\right)$$

This DOF is associated with face 1 of the reference element.
$$\displaystyle l_{11}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(t_{1} \left(2 t_{1} - 1\right))\hat{\boldsymbol{n}}_{1}$$

$$\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle \frac{x \left(- 20 x^{2} + 18 x z + 21 x - 18 z - 1\right)}{2}\\\displaystyle 30 x^{2} y - 30 x^{2} - 36 x y z - 18 x y + 36 x z + 18 x - 3 y^{2} + 30 y z^{2} - 6 y z + 5 y - 30 z^{2} + 6 z - 2\\\displaystyle \frac{z \left(18 x z - 18 x - 20 z^{2} + 21 z - 1\right)}{2}\end{array}\right)$$

This DOF is associated with face 1 of the reference element.
$$\displaystyle l_{12}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(2 t_{0}^{2} + 4 t_{0} t_{1} - 3 t_{0} + 2 t_{1}^{2} - 3 t_{1} + 1)\hat{\boldsymbol{n}}_{2}$$

$$\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 3 x^{2} - 30 x y^{2} - 36 x y z + 54 x y - 30 x z^{2} + 54 x z - 29 x + 30 y^{2} + 36 y z - 54 y + 30 z^{2} - 54 z + 26\\\displaystyle \frac{y \left(20 y^{2} + 18 y z - 39 y - 18 z + 19\right)}{2}\\\displaystyle \frac{z \left(18 y z - 18 y + 20 z^{2} - 39 z + 19\right)}{2}\end{array}\right)$$

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(4 t_{0} \left(- t_{0} - t_{1} + 1\right))\hat{\boldsymbol{n}}_{2}$$

$$\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 3 x^{2} + 15 x y^{2} - 15 x y - 30 x z^{2} + 36 x z - \frac{19 x}{2} - 15 y^{2} + 15 y + 30 z^{2} - 36 z + \frac{13}{2}\\\displaystyle \frac{5 y \left(- 2 y^{2} + 3 y - 1\right)}{2}\\\displaystyle 5 z \left(2 z^{2} - 3 z + 1\right)\end{array}\right)$$

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{14}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(t_{0} \left(2 t_{0} - 1\right))\hat{\boldsymbol{n}}_{2}$$

$$\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 3 x^{2} - 30 x y^{2} + 36 x y z + 6 x y - 30 x z^{2} + 18 x z - 5 x + 30 y^{2} - 36 y z - 6 y + 30 z^{2} - 18 z + 2\\\displaystyle \frac{y \left(20 y^{2} - 18 y z - 21 y + 18 z + 1\right)}{2}\\\displaystyle \frac{z \left(- 18 y z + 18 y + 20 z^{2} - 21 z + 1\right)}{2}\end{array}\right)$$

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{15}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(4 t_{1} \left(- t_{0} - t_{1} + 1\right))\hat{\boldsymbol{n}}_{2}$$

$$\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 3 x^{2} - 30 x y^{2} + 36 x y + 15 x z^{2} - 15 x z - \frac{19 x}{2} + 30 y^{2} - 36 y - 15 z^{2} + 15 z + \frac{13}{2}\\\displaystyle 5 y \left(2 y^{2} - 3 y + 1\right)\\\displaystyle \frac{5 z \left(- 2 z^{2} + 3 z - 1\right)}{2}\end{array}\right)$$

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{16}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(4 t_{0} t_{1})\hat{\boldsymbol{n}}_{2}$$

$$\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle 3 x^{2} + 15 x y^{2} - 15 x y + 15 x z^{2} - 15 x z + x - 15 y^{2} + 15 y - 15 z^{2} + 15 z - 4\\\displaystyle \frac{5 y \left(- 2 y^{2} + 3 y - 1\right)}{2}\\\displaystyle \frac{5 z \left(- 2 z^{2} + 3 z - 1\right)}{2}\end{array}\right)$$

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{17}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(t_{1} \left(2 t_{1} - 1\right))\hat{\boldsymbol{n}}_{2}$$

$$\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle 3 x^{2} - 30 x y^{2} + 36 x y z + 18 x y - 30 x z^{2} + 6 x z - 5 x + 30 y^{2} - 36 y z - 18 y + 30 z^{2} - 6 z + 2\\\displaystyle \frac{y \left(20 y^{2} - 18 y z - 21 y + 18 z + 1\right)}{2}\\\displaystyle \frac{z \left(- 18 y z + 18 y + 20 z^{2} - 21 z + 1\right)}{2}\end{array}\right)$$

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{18}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(2 t_{0}^{2} + 4 t_{0} t_{1} - 3 t_{0} + 2 t_{1}^{2} - 3 t_{1} + 1)\hat{\boldsymbol{n}}_{3}$$

$$\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle x \left(3 x + 30 y^{2} + 36 y z - 54 y + 30 z^{2} - 54 z + 23\right)\\\displaystyle \frac{y \left(- 20 y^{2} - 18 y z + 39 y + 18 z - 19\right)}{2}\\\displaystyle \frac{z \left(- 18 y z + 18 y - 20 z^{2} + 39 z - 19\right)}{2}\end{array}\right)$$

This DOF is associated with face 3 of the reference element.
$$\displaystyle l_{19}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(4 t_{0} \left(- t_{0} - t_{1} + 1\right))\hat{\boldsymbol{n}}_{3}$$

$$\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle \frac{x \left(6 x - 30 y^{2} + 30 y + 60 z^{2} - 72 z + 7\right)}{2}\\\displaystyle \frac{5 y \left(2 y^{2} - 3 y + 1\right)}{2}\\\displaystyle 5 z \left(- 2 z^{2} + 3 z - 1\right)\end{array}\right)$$

This DOF is associated with face 3 of the reference element.
$$\displaystyle l_{20}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(t_{0} \left(2 t_{0} - 1\right))\hat{\boldsymbol{n}}_{3}$$

$$\displaystyle \boldsymbol{\phi}_{20} = \left(\begin{array}{c}\displaystyle x \left(3 x + 30 y^{2} - 36 y z - 6 y + 30 z^{2} - 18 z - 1\right)\\\displaystyle \frac{y \left(- 20 y^{2} + 18 y z + 21 y - 18 z - 1\right)}{2}\\\displaystyle \frac{z \left(18 y z - 18 y - 20 z^{2} + 21 z - 1\right)}{2}\end{array}\right)$$

This DOF is associated with face 3 of the reference element.
$$\displaystyle l_{21}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(4 t_{1} \left(- t_{0} - t_{1} + 1\right))\hat{\boldsymbol{n}}_{3}$$

$$\displaystyle \boldsymbol{\phi}_{21} = \left(\begin{array}{c}\displaystyle \frac{x \left(6 x + 60 y^{2} - 72 y - 30 z^{2} + 30 z + 7\right)}{2}\\\displaystyle 5 y \left(- 2 y^{2} + 3 y - 1\right)\\\displaystyle \frac{5 z \left(2 z^{2} - 3 z + 1\right)}{2}\end{array}\right)$$

This DOF is associated with face 3 of the reference element.
$$\displaystyle l_{22}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(4 t_{0} t_{1})\hat{\boldsymbol{n}}_{3}$$

$$\displaystyle \boldsymbol{\phi}_{22} = \left(\begin{array}{c}\displaystyle x \left(3 x - 15 y^{2} + 15 y - 15 z^{2} + 15 z - 7\right)\\\displaystyle \frac{5 y \left(2 y^{2} - 3 y + 1\right)}{2}\\\displaystyle \frac{5 z \left(2 z^{2} - 3 z + 1\right)}{2}\end{array}\right)$$

This DOF is associated with face 3 of the reference element.
$$\displaystyle l_{23}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(t_{1} \left(2 t_{1} - 1\right))\hat{\boldsymbol{n}}_{3}$$

$$\displaystyle \boldsymbol{\phi}_{23} = \left(\begin{array}{c}\displaystyle x \left(3 x + 30 y^{2} - 36 y z - 18 y + 30 z^{2} - 6 z - 1\right)\\\displaystyle \frac{y \left(- 20 y^{2} + 18 y z + 21 y - 18 z - 1\right)}{2}\\\displaystyle \frac{z \left(18 y z - 18 y - 20 z^{2} + 21 z - 1\right)}{2}\end{array}\right)$$

This DOF is associated with face 3 of the reference element.
$$\displaystyle l_{24}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot(2 t_{0}^{2} + 4 t_{0} t_{1} - 3 t_{0} + 2 t_{1}^{2} - 3 t_{1} + 1)\hat{\boldsymbol{n}}_{4}$$

$$\displaystyle \boldsymbol{\phi}_{24} = \left(\begin{array}{c}\displaystyle \frac{x \left(20 x^{2} + 18 x z - 39 x - 18 z + 19\right)}{2}\\\displaystyle y \left(- 30 x^{2} - 36 x z + 54 x - 3 y - 30 z^{2} + 54 z - 23\right)\\\displaystyle \frac{z \left(18 x z - 18 x + 20 z^{2} - 39 z + 19\right)}{2}\end{array}\right)$$

This DOF is associated with face 4 of the reference element.
$$\displaystyle l_{25}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot(4 t_{0} \left(- t_{0} - t_{1} + 1\right))\hat{\boldsymbol{n}}_{4}$$

$$\displaystyle \boldsymbol{\phi}_{25} = \left(\begin{array}{c}\displaystyle \frac{5 x \left(- 2 x^{2} + 3 x - 1\right)}{2}\\\displaystyle \frac{y \left(30 x^{2} - 30 x - 6 y - 60 z^{2} + 72 z - 7\right)}{2}\\\displaystyle 5 z \left(2 z^{2} - 3 z + 1\right)\end{array}\right)$$

This DOF is associated with face 4 of the reference element.
$$\displaystyle l_{26}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot(t_{0} \left(2 t_{0} - 1\right))\hat{\boldsymbol{n}}_{4}$$

$$\displaystyle \boldsymbol{\phi}_{26} = \left(\begin{array}{c}\displaystyle \frac{x \left(20 x^{2} - 18 x z - 21 x + 18 z + 1\right)}{2}\\\displaystyle y \left(- 30 x^{2} + 36 x z + 6 x - 3 y - 30 z^{2} + 18 z + 1\right)\\\displaystyle \frac{z \left(- 18 x z + 18 x + 20 z^{2} - 21 z + 1\right)}{2}\end{array}\right)$$

This DOF is associated with face 4 of the reference element.
$$\displaystyle l_{27}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot(4 t_{1} \left(- t_{0} - t_{1} + 1\right))\hat{\boldsymbol{n}}_{4}$$

$$\displaystyle \boldsymbol{\phi}_{27} = \left(\begin{array}{c}\displaystyle 5 x \left(2 x^{2} - 3 x + 1\right)\\\displaystyle \frac{y \left(- 60 x^{2} + 72 x - 6 y + 30 z^{2} - 30 z - 7\right)}{2}\\\displaystyle \frac{5 z \left(- 2 z^{2} + 3 z - 1\right)}{2}\end{array}\right)$$

This DOF is associated with face 4 of the reference element.
$$\displaystyle l_{28}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot(4 t_{0} t_{1})\hat{\boldsymbol{n}}_{4}$$

$$\displaystyle \boldsymbol{\phi}_{28} = \left(\begin{array}{c}\displaystyle \frac{5 x \left(- 2 x^{2} + 3 x - 1\right)}{2}\\\displaystyle y \left(15 x^{2} - 15 x - 3 y + 15 z^{2} - 15 z + 7\right)\\\displaystyle \frac{5 z \left(- 2 z^{2} + 3 z - 1\right)}{2}\end{array}\right)$$

This DOF is associated with face 4 of the reference element.
$$\displaystyle l_{29}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot(t_{1} \left(2 t_{1} - 1\right))\hat{\boldsymbol{n}}_{4}$$

$$\displaystyle \boldsymbol{\phi}_{29} = \left(\begin{array}{c}\displaystyle \frac{x \left(20 x^{2} - 18 x z - 21 x + 18 z + 1\right)}{2}\\\displaystyle y \left(- 30 x^{2} + 36 x z + 18 x - 3 y - 30 z^{2} + 6 z + 1\right)\\\displaystyle \frac{z \left(- 18 x z + 18 x + 20 z^{2} - 21 z + 1\right)}{2}\end{array}\right)$$

This DOF is associated with face 4 of the reference element.
$$\displaystyle l_{30}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot(2 t_{0}^{2} + 4 t_{0} t_{1} - 3 t_{0} + 2 t_{1}^{2} - 3 t_{1} + 1)\hat{\boldsymbol{n}}_{5}$$

$$\displaystyle \boldsymbol{\phi}_{30} = \left(\begin{array}{c}\displaystyle \frac{x \left(- 20 x^{2} - 18 x y + 39 x + 18 y - 19\right)}{2}\\\displaystyle \frac{y \left(- 18 x y + 18 x - 20 y^{2} + 39 y - 19\right)}{2}\\\displaystyle z \left(30 x^{2} + 36 x y - 54 x + 30 y^{2} - 54 y + 3 z + 23\right)\end{array}\right)$$

This DOF is associated with face 5 of the reference element.
$$\displaystyle l_{31}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot(4 t_{0} \left(- t_{0} - t_{1} + 1\right))\hat{\boldsymbol{n}}_{5}$$

$$\displaystyle \boldsymbol{\phi}_{31} = \left(\begin{array}{c}\displaystyle \frac{5 x \left(2 x^{2} - 3 x + 1\right)}{2}\\\displaystyle 5 y \left(- 2 y^{2} + 3 y - 1\right)\\\displaystyle \frac{z \left(- 30 x^{2} + 30 x + 60 y^{2} - 72 y + 6 z + 7\right)}{2}\end{array}\right)$$

This DOF is associated with face 5 of the reference element.
$$\displaystyle l_{32}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot(t_{0} \left(2 t_{0} - 1\right))\hat{\boldsymbol{n}}_{5}$$

$$\displaystyle \boldsymbol{\phi}_{32} = \left(\begin{array}{c}\displaystyle \frac{x \left(- 20 x^{2} + 18 x y + 21 x - 18 y - 1\right)}{2}\\\displaystyle \frac{y \left(18 x y - 18 x - 20 y^{2} + 21 y - 1\right)}{2}\\\displaystyle z \left(30 x^{2} - 36 x y - 6 x + 30 y^{2} - 18 y + 3 z - 1\right)\end{array}\right)$$

This DOF is associated with face 5 of the reference element.
$$\displaystyle l_{33}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot(4 t_{1} \left(- t_{0} - t_{1} + 1\right))\hat{\boldsymbol{n}}_{5}$$

$$\displaystyle \boldsymbol{\phi}_{33} = \left(\begin{array}{c}\displaystyle 5 x \left(- 2 x^{2} + 3 x - 1\right)\\\displaystyle \frac{5 y \left(2 y^{2} - 3 y + 1\right)}{2}\\\displaystyle \frac{z \left(60 x^{2} - 72 x - 30 y^{2} + 30 y + 6 z + 7\right)}{2}\end{array}\right)$$

This DOF is associated with face 5 of the reference element.
$$\displaystyle l_{34}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot(4 t_{0} t_{1})\hat{\boldsymbol{n}}_{5}$$

$$\displaystyle \boldsymbol{\phi}_{34} = \left(\begin{array}{c}\displaystyle \frac{5 x \left(2 x^{2} - 3 x + 1\right)}{2}\\\displaystyle \frac{5 y \left(2 y^{2} - 3 y + 1\right)}{2}\\\displaystyle z \left(- 15 x^{2} + 15 x - 15 y^{2} + 15 y + 3 z - 7\right)\end{array}\right)$$

This DOF is associated with face 5 of the reference element.
$$\displaystyle l_{35}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot(t_{1} \left(2 t_{1} - 1\right))\hat{\boldsymbol{n}}_{5}$$

$$\displaystyle \boldsymbol{\phi}_{35} = \left(\begin{array}{c}\displaystyle \frac{x \left(- 20 x^{2} + 18 x y + 21 x - 18 y - 1\right)}{2}\\\displaystyle \frac{y \left(18 x y - 18 x - 20 y^{2} + 21 y - 1\right)}{2}\\\displaystyle z \left(30 x^{2} - 36 x y - 18 x + 30 y^{2} - 6 y + 3 z - 1\right)\end{array}\right)$$

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

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

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

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

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

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

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

## References

• Arnold, D. N. and Awanou, G. Finite element differential forms on cubical meshes, Mathematics of computation 83, 1551–5170, 2014. [BibTeX]
• Brezzi, F., Douglas, J., and Marini, L. D. Two families of mixed finite elements for second order elliptic problems, Numerische Mathematik 47, 217–235, 1985. [DOI: 10.1007/BF01389710] [BibTeX]