an encyclopedia of finite element definitions

# Brezzi–Douglas–Marini

 Abbreviated names BDM Orders $$1\leqslant k$$ Reference elements triangle, tetrahedron Polynomial set $$\mathcal{P}_{k}^d$$↓ Show polynomial set definitions ↓ DOFs On each facet: normal integral moments with an order $$k$$ Lagrange space On the interior of the reference element: integral moments with an order $$k-1$$ Nédélec (first kind) space Number of DOFs triangle: $$(k+1)(k+2)$$ (A002378)tetrahedron: $$(k+1)(k+2)(k+3)/2$$ (A027480) Categories Vector-valued elements, H(div) conforming elements

## Implementations

 Symfem string "N2div"↓ Show Symfem examples ↓ Basix string "Brezzi-Douglas-Marini"↓ Show Basix examples ↓ UFL string "BDM"↓ Show UFL examples ↓

## Examples

triangle
order 1
triangle
order 2
tetrahedron
order 1
tetrahedron
order 2
• $$R$$ is the reference triangle. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{5}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(1 - t_{0})\hat{\boldsymbol{n}}_{0}$$

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

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

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

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

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

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

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

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

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

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

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

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

$$\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 9 x \left(1 - 2 x\right)\\\displaystyle 3 y \left(4 x - 1\right)\end{array}\right)$$

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

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

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

$$\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 3 x \left(4 y - 1\right)\\\displaystyle 9 y \left(1 - 2 y\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle - 18 x^{2} - 48 x y + 27 x - 30 y^{2} + 36 y - 9\\\displaystyle 3 y \left(4 x + 4 y - 3\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle - \frac{3 x^{2}}{2} + 15 x y + 15 y^{2} - 15 y + \frac{3}{2}\\\displaystyle \frac{3 y \left(- 6 x - 5 y + 4\right)}{2}\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle - 12 x y + 3 x - 30 y^{2} + 24 y - 3\\\displaystyle 9 y \left(2 y - 1\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 3 x \left(- 4 x - 4 y + 3\right)\\\displaystyle 30 x^{2} + 48 x y - 36 x + 18 y^{2} - 27 y + 9\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle \frac{3 x \left(5 x + 6 y - 4\right)}{2}\\\displaystyle - 15 x^{2} - 15 x y + 15 x + \frac{3 y^{2}}{2} - \frac{3}{2}\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 9 x \left(1 - 2 x\right)\\\displaystyle 30 x^{2} + 12 x y - 24 x - 3 y + 3\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{9}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}- t_{1}\\t_{0}\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 12 x \left(- x - 4 y + 1\right)\\\displaystyle 12 y \left(4 x + y - 1\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{10}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}t_{1}\\1 - t_{0}\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 12 x \left(x + 2 y - 1\right)\\\displaystyle 12 y \left(- 4 x - 3 y + 3\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 12 x \left(- 3 x - 4 y + 3\right)\\\displaystyle 12 y \left(2 x + y - 1\right)\end{array}\right)$$

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

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

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

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

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

$$\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle - 6 x\\\displaystyle - 6 y\\\displaystyle 18 z\end{array}\right)$$

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

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

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

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

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

$$\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 6 x + 24 z - 6\\\displaystyle 6 y\\\displaystyle - 18 z\end{array}\right)$$

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

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

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

$$\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 18 x\\\displaystyle - 24 x - 6 y + 6\\\displaystyle - 6 z\end{array}\right)$$

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

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

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

$$\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 6 x\\\displaystyle 6 y\\\displaystyle - 24 x - 24 y - 18 z + 18\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle - 18 x\\\displaystyle 6 y\\\displaystyle 24 x + 6 z - 6\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 6 x\\\displaystyle - 18 y\\\displaystyle 24 y + 6 z - 6\end{array}\right)$$

This DOF is associated with face 3 of the reference element.
• $$R$$ is the reference tetrahedron. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle z^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle z^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z^{2}\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{29}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(2 t_{0}^{2} + 4 t_{0} t_{1} - 3 t_{0} + 2 t_{1}^{2} - 3 t_{1} + 1)\hat{\boldsymbol{n}}_{0}$$

$$\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 24 x \left(5 x - 2\right)\\\displaystyle 12 y \left(1 - 5 x\right)\\\displaystyle 12 z \left(1 - 5 x\right)\end{array}\right)$$

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

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

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

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

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

$$\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 3 x \left(5 x + 20 z - 6\right)\\\displaystyle 6 y \left(- 5 x - 5 z + 2\right)\\\displaystyle 3 z \left(20 x + 5 z - 6\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 6 x \left(- 5 y - 5 z + 2\right)\\\displaystyle 3 y \left(5 y + 20 z - 6\right)\\\displaystyle 3 z \left(20 y + 5 z - 6\right)\end{array}\right)$$

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

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

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

$$\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 120 x^{2} + 300 x y + 300 x z - 192 x + 180 y^{2} + 360 y z - 240 y + 180 z^{2} - 240 z + 72\\\displaystyle 12 y \left(- 5 x - 5 y - 5 z + 4\right)\\\displaystyle 12 z \left(- 5 x - 5 y - 5 z + 4\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 15 x^{2} - 90 x y + 60 x z - 12 x - 90 y^{2} - 90 y z + 90 y + 45 z^{2} - 30 z - 3\\\displaystyle 3 y \left(20 x + 15 y + 20 z - 14\right)\\\displaystyle 6 z \left(- 5 x - 5 z + 3\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 60 x y - 12 x + 180 y^{2} - 120 y + 12\\\displaystyle 24 y \left(2 - 5 y\right)\\\displaystyle 12 z \left(5 y - 1\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 15 x^{2} + 60 x y - 90 x z - 12 x + 45 y^{2} - 90 y z - 30 y - 90 z^{2} + 90 z - 3\\\displaystyle 6 y \left(- 5 x - 5 y + 3\right)\\\displaystyle 3 z \left(20 x + 20 y + 15 z - 14\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 30 x y + 30 x z - 12 x + 45 y^{2} + 180 y z - 60 y + 45 z^{2} - 60 z + 12\\\displaystyle 3 y \left(- 5 y - 20 z + 6\right)\\\displaystyle 3 z \left(- 20 y - 5 z + 6\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 60 x z - 12 x + 180 z^{2} - 120 z + 12\\\displaystyle 12 y \left(5 z - 1\right)\\\displaystyle 24 z \left(2 - 5 z\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 12 x \left(5 x + 5 y + 5 z - 4\right)\\\displaystyle - 180 x^{2} - 300 x y - 360 x z + 240 x - 120 y^{2} - 300 y z + 192 y - 180 z^{2} + 240 z - 72\\\displaystyle 12 z \left(5 x + 5 y + 5 z - 4\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 3 x \left(- 15 x - 20 y - 20 z + 14\right)\\\displaystyle 90 x^{2} + 90 x y + 90 x z - 90 x - 15 y^{2} - 60 y z + 12 y - 45 z^{2} + 30 z + 3\\\displaystyle 6 z \left(5 y + 5 z - 3\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 24 x \left(5 x - 2\right)\\\displaystyle - 180 x^{2} - 60 x y + 120 x + 12 y - 12\\\displaystyle 12 z \left(1 - 5 x\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 6 x \left(5 x + 5 y - 3\right)\\\displaystyle - 45 x^{2} - 60 x y + 90 x z + 30 x - 15 y^{2} + 90 y z + 12 y + 90 z^{2} - 90 z + 3\\\displaystyle 3 z \left(- 20 x - 20 y - 15 z + 14\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle 3 x \left(5 x + 20 z - 6\right)\\\displaystyle - 45 x^{2} - 30 x y - 180 x z + 60 x - 30 y z + 12 y - 45 z^{2} + 60 z - 12\\\displaystyle 3 z \left(20 x + 5 z - 6\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle 12 x \left(1 - 5 z\right)\\\displaystyle - 60 y z + 12 y - 180 z^{2} + 120 z - 12\\\displaystyle 24 z \left(5 z - 2\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle 12 x \left(- 5 x - 5 y - 5 z + 4\right)\\\displaystyle 12 y \left(- 5 x - 5 y - 5 z + 4\right)\\\displaystyle 180 x^{2} + 360 x y + 300 x z - 240 x + 180 y^{2} + 300 y z - 240 y + 120 z^{2} - 192 z + 72\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle 3 x \left(15 x + 20 y + 20 z - 14\right)\\\displaystyle 6 y \left(- 5 y - 5 z + 3\right)\\\displaystyle - 90 x^{2} - 90 x y - 90 x z + 90 x + 45 y^{2} + 60 y z - 30 y + 15 z^{2} - 12 z - 3\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{20} = \left(\begin{array}{c}\displaystyle 24 x \left(2 - 5 x\right)\\\displaystyle 12 y \left(5 x - 1\right)\\\displaystyle 180 x^{2} + 60 x z - 120 x - 12 z + 12\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{21} = \left(\begin{array}{c}\displaystyle 6 x \left(- 5 x - 5 z + 3\right)\\\displaystyle 3 y \left(20 x + 15 y + 20 z - 14\right)\\\displaystyle 45 x^{2} - 90 x y + 60 x z - 30 x - 90 y^{2} - 90 y z + 90 y + 15 z^{2} - 12 z - 3\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{22} = \left(\begin{array}{c}\displaystyle 3 x \left(- 5 x - 20 y + 6\right)\\\displaystyle 3 y \left(- 20 x - 5 y + 6\right)\\\displaystyle 45 x^{2} + 180 x y + 30 x z - 60 x + 45 y^{2} + 30 y z - 60 y - 12 z + 12\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{23} = \left(\begin{array}{c}\displaystyle 12 x \left(5 y - 1\right)\\\displaystyle 24 y \left(2 - 5 y\right)\\\displaystyle 180 y^{2} + 60 y z - 120 y - 12 z + 12\end{array}\right)$$

This DOF is associated with face 3 of the reference element.
$$\displaystyle l_{24}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\- t_{2}\\t_{1}\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{24} = \left(\begin{array}{c}\displaystyle 60 x \left(- y + z\right)\\\displaystyle 60 y \left(- y - 5 z + 1\right)\\\displaystyle 60 z \left(5 y + z - 1\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{25} = \left(\begin{array}{c}\displaystyle 60 x \left(- x - 5 z + 1\right)\\\displaystyle 60 y \left(- x + z\right)\\\displaystyle 60 z \left(5 x + z - 1\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{26} = \left(\begin{array}{c}\displaystyle 60 x \left(- x - 5 y + 1\right)\\\displaystyle 60 y \left(5 x + y - 1\right)\\\displaystyle 60 z \left(- x + y\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{27} = \left(\begin{array}{c}\displaystyle 60 x \left(x + y + 2 z - 1\right)\\\displaystyle 60 y \left(x + y + 2 z - 1\right)\\\displaystyle 60 z \left(- 5 x - 5 y - 4 z + 4\right)\end{array}\right)$$

This DOF is associated with volume 0 of the reference element.
$$\displaystyle l_{28}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}t_{1}\\- t_{0} - t_{2} + 1\\t_{1}\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{28} = \left(\begin{array}{c}\displaystyle 60 x \left(x + 2 y + z - 1\right)\\\displaystyle 60 y \left(- 5 x - 4 y - 5 z + 4\right)\\\displaystyle 60 z \left(x + 2 y + z - 1\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{29} = \left(\begin{array}{c}\displaystyle 60 x \left(- 4 x - 5 y - 5 z + 4\right)\\\displaystyle 60 y \left(2 x + y + z - 1\right)\\\displaystyle 60 z \left(2 x + y + z - 1\right)\end{array}\right)$$

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

## References

• 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]