an encyclopedia of finite element definitions

# Q H(div)

 Alternative names Raviart–Thomas cubical H(div) (quadrilateral), Nédélec cubical H(div) (hexahedron), $$\mathcal{Q}^-_k\Lambda^{d-1}(\square_d)$$ Abbreviated names RTcf (quadrilateral), Ncf (hexahedron) Orders $$1\leqslant k$$ Reference elements quadrilateral, hexahedron Polynomial set $$\mathcal{Q}_{k-1}^d \oplus \mathcal{Z}^{(18)}_{k}$$ (quadrilateral) $$\mathcal{Q}_{k}^d \oplus \mathcal{Z}^{(18)}_{k}$$ (hexahedron) ↓ Show polynomial set definitions ↓ DOFs On each facet: normal integral moments with an order $$k-1$$ Lagrange space On the interior of the reference element: integral moments with an order $$k-1$$ Q H(curl) space Number of DOFs quadrilateral: $$2k(k+1)$$ (A046092)hexahedron: $$3k^2(k+1)$$ (A270205) Categories Vector-valued elements, H(div) conforming elements

## Implementations

 Symfem string "Qdiv"↓ Show Symfem examples ↓ UFL string "RTCF" (quadrilateral)"NCF" (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: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{3}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{0}$$

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

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

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

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

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

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

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

This DOF is associated with edge 3 of the reference element.
• $$R$$ is the reference quadrilateral. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2} y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y^{2}\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{11}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(1 - t_{0})\hat{\boldsymbol{n}}_{0}$$

$$\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - 18 x y^{2} + 24 x y - 6 x + 12 y^{2} - 16 y + 4\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 18 x y^{2} - 24 x y + 6 x - 6 y^{2} + 8 y - 2\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 18 x^{2} y - 12 x^{2} - 24 x y + 16 x + 6 y - 4\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle - 18 x^{2} y + 6 x^{2} + 24 x y - 8 x - 6 y + 2\\\displaystyle 0\end{array}\right)$$

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

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

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

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

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

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

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

$$\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 y \left(9 x y - 6 x - 3 y + 2\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 12 x \left(3 x y - 2 x - 3 y + 2\right)\\\displaystyle 0\end{array}\right)$$

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

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

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

$$\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 12 y \left(- 3 x y + 3 x + y - 1\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 12 x \left(- 3 x y + x + 3 y - 1\right)\\\displaystyle 0\end{array}\right)$$

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: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{5}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{0}$$

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

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

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

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

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

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

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

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

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

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

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

This DOF is associated with face 5 of the reference element.
• $$R$$ is the reference hexahedron. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2} z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2} y\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2} y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2} z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y^{2} z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y z^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x z^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y z^{2}\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{35}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(t_{0} t_{1} - t_{0} - t_{1} + 1)\hat{\boldsymbol{n}}_{0}$$

$$\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 108 x y z^{2} - 144 x y z + 36 x y - 72 x z^{2} + 96 x z - 24 x - 72 y z^{2} + 96 y z - 24 y + 48 z^{2} - 64 z + 16\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - 108 x y z^{2} + 144 x y z - 36 x y + 72 x z^{2} - 96 x z + 24 x + 36 y z^{2} - 48 y z + 12 y - 24 z^{2} + 32 z - 8\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - 108 x y z^{2} + 144 x y z - 36 x y + 36 x z^{2} - 48 x z + 12 x + 72 y z^{2} - 96 y z + 24 y - 24 z^{2} + 32 z - 8\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 108 x y z^{2} - 144 x y z + 36 x y - 36 x z^{2} + 48 x z - 12 x - 36 y z^{2} + 48 y z - 12 y + 12 z^{2} - 16 z + 4\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - 108 x y^{2} z + 72 x y^{2} + 144 x y z - 96 x y - 36 x z + 24 x + 72 y^{2} z - 48 y^{2} - 96 y z + 64 y + 24 z - 16\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 108 x y^{2} z - 72 x y^{2} - 144 x y z + 96 x y + 36 x z - 24 x - 36 y^{2} z + 24 y^{2} + 48 y z - 32 y - 12 z + 8\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 108 x y^{2} z - 36 x y^{2} - 144 x y z + 48 x y + 36 x z - 12 x - 72 y^{2} z + 24 y^{2} + 96 y z - 32 y - 24 z + 8\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - 108 x y^{2} z + 36 x y^{2} + 144 x y z - 48 x y - 36 x z + 12 x + 36 y^{2} z - 12 y^{2} - 48 y z + 16 y + 12 z - 4\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 108 x^{2} y z - 72 x^{2} y - 72 x^{2} z + 48 x^{2} - 144 x y z + 96 x y + 96 x z - 64 x + 36 y z - 24 y - 24 z + 16\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle - 108 x^{2} y z + 72 x^{2} y + 36 x^{2} z - 24 x^{2} + 144 x y z - 96 x y - 48 x z + 32 x - 36 y z + 24 y + 12 z - 8\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{10}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(t_{1} \left(1 - t_{0}\right))\hat{\boldsymbol{n}}_{2}$$

$$\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle - 108 x^{2} y z + 36 x^{2} y + 72 x^{2} z - 24 x^{2} + 144 x y z - 48 x y - 96 x z + 32 x - 36 y z + 12 y + 24 z - 8\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 108 x^{2} y z - 36 x^{2} y - 36 x^{2} z + 12 x^{2} - 144 x y z + 48 x y + 48 x z - 16 x + 36 y z - 12 y - 12 z + 4\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 4 x \left(27 x y z - 18 x y - 18 x z + 12 x - 18 y z + 12 y + 12 z - 8\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 3 of the reference element.
$$\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(t_{0} \left(1 - t_{1}\right))\hat{\boldsymbol{n}}_{3}$$

$$\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 4 x \left(- 27 x y z + 18 x y + 9 x z - 6 x + 18 y z - 12 y - 6 z + 4\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 3 of the reference element.
$$\displaystyle l_{14}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(t_{1} \left(1 - t_{0}\right))\hat{\boldsymbol{n}}_{3}$$

$$\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 4 x \left(- 27 x y z + 9 x y + 18 x z - 6 x + 18 y z - 6 y - 12 z + 4\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 3 of the reference element.
$$\displaystyle l_{15}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(t_{0} t_{1})\hat{\boldsymbol{n}}_{3}$$

$$\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 4 x \left(27 x y z - 9 x y - 9 x z + 3 x - 18 y z + 6 y + 6 z - 2\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 3 of the reference element.
$$\displaystyle l_{16}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot(t_{0} t_{1} - t_{0} - t_{1} + 1)\hat{\boldsymbol{n}}_{4}$$

$$\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 y \left(- 27 x y z + 18 x y + 18 x z - 12 x + 18 y z - 12 y - 12 z + 8\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 4 of the reference element.
$$\displaystyle l_{17}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot(t_{0} \left(1 - t_{1}\right))\hat{\boldsymbol{n}}_{4}$$

$$\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 y \left(27 x y z - 18 x y - 18 x z + 12 x - 9 y z + 6 y + 6 z - 4\right)\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 y \left(27 x y z - 9 x y - 18 x z + 6 x - 18 y z + 6 y + 12 z - 4\right)\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 y \left(- 27 x y z + 9 x y + 18 x z - 6 x + 9 y z - 3 y - 6 z + 2\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 4 of the reference element.
$$\displaystyle l_{20}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot(t_{0} t_{1} - t_{0} - t_{1} + 1)\hat{\boldsymbol{n}}_{5}$$

$$\displaystyle \boldsymbol{\phi}_{20} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 z \left(27 x y z - 18 x y - 18 x z + 12 x - 18 y z + 12 y + 12 z - 8\right)\end{array}\right)$$

This DOF is associated with face 5 of the reference element.
$$\displaystyle l_{21}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot(t_{0} \left(1 - t_{1}\right))\hat{\boldsymbol{n}}_{5}$$

$$\displaystyle \boldsymbol{\phi}_{21} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 z \left(- 27 x y z + 18 x y + 18 x z - 12 x + 9 y z - 6 y - 6 z + 4\right)\end{array}\right)$$

This DOF is associated with face 5 of the reference element.
$$\displaystyle l_{22}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot(t_{1} \left(1 - t_{0}\right))\hat{\boldsymbol{n}}_{5}$$

$$\displaystyle \boldsymbol{\phi}_{22} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 z \left(- 27 x y z + 18 x y + 9 x z - 6 x + 18 y z - 12 y - 6 z + 4\right)\end{array}\right)$$

This DOF is associated with face 5 of the reference element.
$$\displaystyle l_{23}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot(t_{0} t_{1})\hat{\boldsymbol{n}}_{5}$$

$$\displaystyle \boldsymbol{\phi}_{23} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 z \left(27 x y z - 18 x y - 9 x z + 6 x - 9 y z + 6 y + 3 z - 2\right)\end{array}\right)$$

This DOF is associated with face 5 of the reference element.
$$\displaystyle l_{24}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}t_{1} t_{2} - t_{1} - t_{2} + 1\\0\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{24} = \left(\begin{array}{c}\displaystyle 24 x \left(- 9 x y z + 6 x y + 6 x z - 4 x + 9 y z - 6 y - 6 z + 4\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

This DOF is associated with volume 0 of the reference element.
$$\displaystyle l_{25}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\t_{0} t_{2} - t_{0} - t_{2} + 1\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{25} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 24 y \left(- 9 x y z + 6 x y + 9 x z - 6 x + 6 y z - 4 y - 6 z + 4\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with volume 0 of the reference element.
$$\displaystyle l_{26}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\0\\t_{0} t_{1} - t_{0} - t_{1} + 1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{26} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 24 z \left(- 9 x y z + 9 x y + 6 x z - 6 x + 6 y z - 6 y - 4 z + 4\right)\end{array}\right)$$

This DOF is associated with volume 0 of the reference element.
$$\displaystyle l_{27}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\t_{0} \left(1 - t_{2}\right)\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{27} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 24 y \left(9 x y z - 6 x y - 9 x z + 6 x - 3 y z + 2 y + 3 z - 2\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with volume 0 of the reference element.
$$\displaystyle l_{28}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\0\\t_{0} \left(1 - t_{1}\right)\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{28} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 24 z \left(9 x y z - 9 x y - 6 x z + 6 x - 3 y z + 3 y + 2 z - 2\right)\end{array}\right)$$

This DOF is associated with volume 0 of the reference element.
$$\displaystyle l_{29}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}t_{1} \left(1 - t_{2}\right)\\0\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{29} = \left(\begin{array}{c}\displaystyle 24 x \left(9 x y z - 6 x y - 3 x z + 2 x - 9 y z + 6 y + 3 z - 2\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

This DOF is associated with volume 0 of the reference element.
$$\displaystyle l_{30}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\0\\t_{1} \left(1 - t_{0}\right)\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{30} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 24 z \left(9 x y z - 9 x y - 3 x z + 3 x - 6 y z + 6 y + 2 z - 2\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{31} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 24 z \left(- 9 x y z + 9 x y + 3 x z - 3 x + 3 y z - 3 y - z + 1\right)\end{array}\right)$$

This DOF is associated with volume 0 of the reference element.
$$\displaystyle l_{32}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}t_{2} \left(1 - t_{1}\right)\\0\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{32} = \left(\begin{array}{c}\displaystyle 24 x \left(9 x y z - 3 x y - 6 x z + 2 x - 9 y z + 3 y + 6 z - 2\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

This DOF is associated with volume 0 of the reference element.
$$\displaystyle l_{33}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\t_{2} \left(1 - t_{0}\right)\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{33} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 24 y \left(9 x y z - 3 x y - 9 x z + 3 x - 6 y z + 2 y + 6 z - 2\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with volume 0 of the reference element.
$$\displaystyle l_{34}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\t_{0} t_{2}\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{34} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 24 y \left(- 9 x y z + 3 x y + 9 x z - 3 x + 3 y z - y - 3 z + 1\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with volume 0 of the reference element.
$$\displaystyle l_{35}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}t_{1} t_{2}\\0\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{35} = \left(\begin{array}{c}\displaystyle 24 x \left(- 9 x y z + 3 x y + 3 x z - x + 9 y z - 3 y - 3 z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

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

## References

• Nédélec, J. C. Mixed finite elements in $$\mathbb{R}^3$$, Numerische Mathematik 35(3), 315–341, 1980. [DOI: 10.1007/BF01396415] [BibTeX]