an encyclopedia of finite element definitions

# Bernardi–Raugel

 Abbreviated names BR Orders $$k=1$$ Reference elements triangle, tetrahedron Polynomial set $$\mathcal{P}_{k}^d \oplus \mathcal{Z}^{(46)}_{k}$$ (triangle) $$\mathcal{P}_{k}^d \oplus \mathcal{Z}^{(47)}_{k}$$ (tetrahedron, $$k=1$$) $$\mathcal{P}_{k}^d \oplus \mathcal{Z}^{(48)}_{k}$$ (tetrahedron, $$k=2$$) ↓ Show polynomial set definitions ↓ DOFs On each edge: (if $$k>1$$) point evaluations in tangential directions at midpoints On each facet: point evaluations in normal directions at vertices,normal integral moments with an order $$k-1$$ Lagrange space, and (if $$k>1$$) point evaluations in normal directions at midpoints of edges On the interior of the reference element: integral moments of the divergence with an order $$0$$ vector Lagrange space Number of DOFs triangle: $$9$$tetrahedron: $$\begin{cases}16&k=1\\37&k=2\end{cases}$$ Categories Vector-valued elements, H(div) conforming elements

## Implementations

 Symfem "Bernardi-Raugel"↓ Show Symfem examples ↓

## Examples

triangle
order 1
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: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2} x y}{2}\\\displaystyle - \frac{\sqrt{2} x y}{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y \left(x + y - 1\right)\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x \left(- x - y + 1\right)\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{8}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0)\cdot\left(\begin{array}{c}-1\\-1\end{array}\right)$$

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

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

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

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

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

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

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

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

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

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{5}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0)\cdot\left(\begin{array}{c}0\\1\end{array}\right)$$

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

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot\hat{\boldsymbol{n}}_{0}$$
where $$e_{0}$$ is the 0th edge;
and $$\hat{\boldsymbol{n}}_{0}$$ is the normal to facet 0.

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

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot\hat{\boldsymbol{n}}_{1}$$
where $$e_{1}$$ is the 1st edge;
and $$\hat{\boldsymbol{n}}_{1}$$ is the normal to facet 1.

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

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot\hat{\boldsymbol{n}}_{2}$$
where $$e_{2}$$ is the 2nd edge;
and $$\hat{\boldsymbol{n}}_{2}$$ is the normal to facet 2.

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

This DOF is associated with edge 2 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: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle \frac{\sqrt{3} x y z}{3}\\\displaystyle \frac{\sqrt{3} x y z}{3}\\\displaystyle \frac{\sqrt{3} x y z}{3}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y z \left(- x - y - z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x z \left(x + y + z - 1\right)\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y \left(- x - y - z + 1\right)\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{15}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,0)\cdot\left(\begin{array}{c}1\\1\\1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle \frac{x \left(- 20 y z + 3\right)}{3}\\\displaystyle - \frac{20 x y z}{3}\\\displaystyle - \frac{20 x y z}{3}\end{array}\right)$$

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

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

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

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

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

$$\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 20 x y z - x + 20 y^{2} z + 20 y z^{2} - 20 y z - y - z + 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle y \left(20 x z + 20 y z + 20 z^{2} - 20 z + 1\right)\\\displaystyle - y\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle z \left(20 x y + 20 y^{2} + 20 y z - 20 y + 1\right)\\\displaystyle 0\\\displaystyle - z\end{array}\right)$$

This DOF is associated with face 1 of the reference element.
$$\displaystyle l_{6}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,0)\cdot\left(\begin{array}{c}0\\-1\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - 20 x^{2} z - 20 x y z - 20 x z^{2} + 20 x z + x + y + z - 1\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{7}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,0)\cdot\left(\begin{array}{c}0\\-1\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle x\\\displaystyle x \left(- 20 x z - 20 y z - 20 z^{2} + 20 z - 1\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{8}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,1)\cdot\left(\begin{array}{c}0\\-1\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle z \left(- 20 x^{2} - 20 x y - 20 x z + 20 x - 1\right)\\\displaystyle z\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 20 x^{2} y + 20 x y^{2} + 20 x y z - 20 x y - x - y - z + 1\end{array}\right)$$

This DOF is associated with face 3 of the reference element.
$$\displaystyle l_{10}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle - x\\\displaystyle 0\\\displaystyle x \left(20 x y + 20 y^{2} + 20 y z - 20 y + 1\right)\end{array}\right)$$

This DOF is associated with face 3 of the reference element.
$$\displaystyle l_{11}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - y\\\displaystyle y \left(20 x^{2} + 20 x y + 20 x z - 20 x + 1\right)\end{array}\right)$$

This DOF is associated with face 3 of the reference element.
$$\displaystyle l_{12}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot\hat{\boldsymbol{n}}_{0}$$
where $$f_{0}$$ is the 0th face;
and $$\hat{\boldsymbol{n}}_{0}$$ is the normal to facet 0.

$$\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 40 x y z\\\displaystyle 40 x y z\\\displaystyle 40 x y z\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot\hat{\boldsymbol{n}}_{1}$$
where $$f_{1}$$ is the 1st face;
and $$\hat{\boldsymbol{n}}_{1}$$ is the normal to facet 1.

$$\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 120 y z \left(- x - y - z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 1 of the reference element.
$$\displaystyle l_{14}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot\hat{\boldsymbol{n}}_{2}$$
where $$f_{2}$$ is the 2nd face;
and $$\hat{\boldsymbol{n}}_{2}$$ is the normal to facet 2.

$$\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 120 x z \left(x + y + z - 1\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{15}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot\hat{\boldsymbol{n}}_{3}$$
where $$f_{3}$$ is the 3th face;
and $$\hat{\boldsymbol{n}}_{3}$$ is the normal to facet 3.

$$\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 120 x y \left(- x - y - z + 1\right)\end{array}\right)$$

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: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle z^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle z^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle \frac{\sqrt{3} x y z}{3}\\\displaystyle \frac{\sqrt{3} x y z}{3}\\\displaystyle \frac{\sqrt{3} x y z}{3}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y z \left(- x - y - z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x z \left(x + y + z - 1\right)\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y \left(- x - y - z + 1\right)\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y z \left(- x - y - z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y z \left(- x - y - z + 1\right)\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y z \left(- x - y - z + 1\right)\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{36}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,0)\cdot\left(\begin{array}{c}1\\1\\1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle x \left(- 126 x y z + 2 x - 126 y^{2} z - 126 y z^{2} + 126 y z - 1\right)\\\displaystyle 42 x y z \left(x + y + z - 1\right)\\\displaystyle 42 x y z \left(x + y + z - 1\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 42 x y z \left(x + y + z - 1\right)\\\displaystyle y \left(- 126 x^{2} z - 126 x y z - 126 x z^{2} + 126 x z + 2 y - 1\right)\\\displaystyle 42 x y z \left(x + y + z - 1\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 42 x y z \left(x + y + z - 1\right)\\\displaystyle 42 x y z \left(x + y + z - 1\right)\\\displaystyle z \left(- 126 x^{2} y - 126 x y^{2} - 126 x y z + 126 x y + 2 z - 1\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle - 42 x^{2} y z + 2 x^{2} - 42 x y^{2} z - 42 x y z^{2} + 42 x y z + 4 x y + 4 x z - 3 x + 2 y^{2} + 4 y z - 3 y + 2 z^{2} - 3 z + 1\\\displaystyle 42 x y z \left(- x - y - z + 1\right)\\\displaystyle 42 x y z \left(- x - y - z + 1\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle y \left(- 42 x^{2} z - 42 x y z - 42 x z^{2} + 42 x z + 2 y - 1\right)\\\displaystyle y \left(126 x^{2} z + 126 x y z + 126 x z^{2} - 126 x z - 2 y + 1\right)\\\displaystyle 42 x y z \left(- x - y - z + 1\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle z \left(- 42 x^{2} y - 42 x y^{2} - 42 x y z + 42 x y + 2 z - 1\right)\\\displaystyle 42 x y z \left(- x - y - z + 1\right)\\\displaystyle z \left(126 x^{2} y + 126 x y^{2} + 126 x y z - 126 x y - 2 z + 1\right)\end{array}\right)$$

This DOF is associated with face 1 of the reference element.
$$\displaystyle l_{6}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,0)\cdot\left(\begin{array}{c}0\\-1\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 42 x y z \left(x + y + z - 1\right)\\\displaystyle 42 x^{2} y z - 2 x^{2} + 42 x y^{2} z + 42 x y z^{2} - 42 x y z - 4 x y - 4 x z + 3 x - 2 y^{2} - 4 y z + 3 y - 2 z^{2} + 3 z - 1\\\displaystyle 42 x y z \left(x + y + z - 1\right)\end{array}\right)$$

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{7}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,0)\cdot\left(\begin{array}{c}0\\-1\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle x \left(- 126 x y z + 2 x - 126 y^{2} z - 126 y z^{2} + 126 y z - 1\right)\\\displaystyle x \left(42 x y z - 2 x + 42 y^{2} z + 42 y z^{2} - 42 y z + 1\right)\\\displaystyle 42 x y z \left(x + y + z - 1\right)\end{array}\right)$$

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{8}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,1)\cdot\left(\begin{array}{c}0\\-1\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 42 x y z \left(x + y + z - 1\right)\\\displaystyle z \left(42 x^{2} y + 42 x y^{2} + 42 x y z - 42 x y - 2 z + 1\right)\\\displaystyle z \left(- 126 x^{2} y - 126 x y^{2} - 126 x y z + 126 x y + 2 z - 1\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 42 x y z \left(- x - y - z + 1\right)\\\displaystyle 42 x y z \left(- x - y - z + 1\right)\\\displaystyle - 42 x^{2} y z + 2 x^{2} - 42 x y^{2} z - 42 x y z^{2} + 42 x y z + 4 x y + 4 x z - 3 x + 2 y^{2} + 4 y z - 3 y + 2 z^{2} - 3 z + 1\end{array}\right)$$

This DOF is associated with face 3 of the reference element.
$$\displaystyle l_{10}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle x \left(126 x y z - 2 x + 126 y^{2} z + 126 y z^{2} - 126 y z + 1\right)\\\displaystyle 42 x y z \left(- x - y - z + 1\right)\\\displaystyle x \left(- 42 x y z + 2 x - 42 y^{2} z - 42 y z^{2} + 42 y z - 1\right)\end{array}\right)$$

This DOF is associated with face 3 of the reference element.
$$\displaystyle l_{11}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 42 x y z \left(- x - y - z + 1\right)\\\displaystyle y \left(126 x^{2} z + 126 x y z + 126 x z^{2} - 126 x z - 2 y + 1\right)\\\displaystyle y \left(- 42 x^{2} z - 42 x y z - 42 x z^{2} + 42 x z + 2 y - 1\right)\end{array}\right)$$

This DOF is associated with face 3 of the reference element.
$$\displaystyle l_{12}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot\hat{\boldsymbol{n}}_{0}$$
where $$f_{0}$$ is the 0th face;
and $$\hat{\boldsymbol{n}}_{0}$$ is the normal to facet 0.

$$\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 40 x y z \left(- 35 x - 35 y - 35 z + 36\right)\\\displaystyle 40 x y z \left(- 35 x - 35 y - 35 z + 36\right)\\\displaystyle 40 x y z \left(- 35 x - 35 y - 35 z + 36\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot\hat{\boldsymbol{n}}_{1}$$
where $$f_{1}$$ is the 1st face;
and $$\hat{\boldsymbol{n}}_{1}$$ is the normal to facet 1.

$$\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 120 y z \left(7 x^{2} + 7 x y + 7 x z - 8 x - y - z + 1\right)\\\displaystyle 1680 x y z \left(x + y + z - 1\right)\\\displaystyle 1680 x y z \left(x + y + z - 1\right)\end{array}\right)$$

This DOF is associated with face 1 of the reference element.
$$\displaystyle l_{14}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot\hat{\boldsymbol{n}}_{2}$$
where $$f_{2}$$ is the 2nd face;
and $$\hat{\boldsymbol{n}}_{2}$$ is the normal to facet 2.

$$\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 1680 x y z \left(- x - y - z + 1\right)\\\displaystyle 120 x z \left(- 7 x y + x - 7 y^{2} - 7 y z + 8 y + z - 1\right)\\\displaystyle 1680 x y z \left(- x - y - z + 1\right)\end{array}\right)$$

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{15}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot\hat{\boldsymbol{n}}_{3}$$
where $$f_{3}$$ is the 3th face;
and $$\hat{\boldsymbol{n}}_{3}$$ is the normal to facet 3.

$$\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 1680 x y z \left(x + y + z - 1\right)\\\displaystyle 1680 x y z \left(x + y + z - 1\right)\\\displaystyle 120 x y \left(7 x z - x + 7 y z - y + 7 z^{2} - 8 z + 1\right)\end{array}\right)$$

This DOF is associated with face 3 of the reference element.
$$\displaystyle l_{16}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{2},\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\-1\\1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 y z \left(- 42 x^{2} - 42 x y - 42 x z + 42 x - 1\right)\\\displaystyle 2 y z \left(42 x^{2} + 42 x y + 42 x z - 42 x + 1\right)\end{array}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{17}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},0,\tfrac{1}{2})\cdot\left(\begin{array}{c}-1\\0\\1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle 2 x z \left(- 42 x y - 42 y^{2} - 42 y z + 42 y - 1\right)\\\displaystyle 0\\\displaystyle 2 x z \left(42 x y + 42 y^{2} + 42 y z - 42 y + 1\right)\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{18}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},\tfrac{1}{2},0)\cdot\left(\begin{array}{c}-1\\1\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle 2 x y \left(- 42 x z - 42 y z - 42 z^{2} + 42 z - 1\right)\\\displaystyle 2 x y \left(42 x z + 42 y z + 42 z^{2} - 42 z + 1\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{19}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 z \left(42 x^{2} y + 42 x y^{2} + 42 x y z - 42 x y - x - y - z + 1\right)\end{array}\right)$$

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{20}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{2},0)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{20} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 y \left(42 x^{2} z + 42 x y z + 42 x z^{2} - 42 x z - x - y - z + 1\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 4 of the reference element.
$$\displaystyle l_{21}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},0,0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{21} = \left(\begin{array}{c}\displaystyle 4 x \left(42 x y z - x + 42 y^{2} z + 42 y z^{2} - 42 y z - y - z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 5 of the reference element.
$$\displaystyle l_{22}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{2},\tfrac{1}{2})\cdot\left(\begin{array}{c}1\\1\\1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{22} = \left(\begin{array}{c}\displaystyle \frac{4 x y z \left(49 x + 49 y + 49 z - 54\right)}{3}\\\displaystyle \frac{2 y z \left(- 28 x^{2} - 28 x y - 28 x z + 18 x + 3\right)}{3}\\\displaystyle \frac{2 y z \left(- 28 x^{2} - 28 x y - 28 x z + 18 x + 3\right)}{3}\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{23}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},0,\tfrac{1}{2})\cdot\left(\begin{array}{c}1\\1\\1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{23} = \left(\begin{array}{c}\displaystyle \frac{2 x z \left(- 28 x y - 28 y^{2} - 28 y z + 18 y + 3\right)}{3}\\\displaystyle \frac{4 x y z \left(49 x + 49 y + 49 z - 54\right)}{3}\\\displaystyle \frac{2 x z \left(- 28 x y - 28 y^{2} - 28 y z + 18 y + 3\right)}{3}\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{24} = \left(\begin{array}{c}\displaystyle \frac{2 x y \left(- 28 x z - 28 y z - 28 z^{2} + 18 z + 3\right)}{3}\\\displaystyle \frac{2 x y \left(- 28 x z - 28 y z - 28 z^{2} + 18 z + 3\right)}{3}\\\displaystyle \frac{4 x y z \left(49 x + 49 y + 49 z - 54\right)}{3}\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{25}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{2},\tfrac{1}{2})\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{25} = \left(\begin{array}{c}\displaystyle 4 y z \left(7 x^{2} + 7 x y + 7 x z - 2 x + 5 y + 5 z - 4\right)\\\displaystyle 2 y z \left(- 14 x^{2} - 14 x y - 14 x z + 14 x - 1\right)\\\displaystyle 2 y z \left(- 14 x^{2} - 14 x y - 14 x z + 14 x - 1\right)\end{array}\right)$$

This DOF is associated with face 1 of the reference element.
$$\displaystyle l_{26}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,\tfrac{1}{2})\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{26} = \left(\begin{array}{c}\displaystyle 4 z \left(7 x^{2} y + 7 x y^{2} + 7 x y z - 2 x y - x + 5 y^{2} + 5 y z - 6 y - z + 1\right)\\\displaystyle 112 x y z \left(- x - y - z + 1\right)\\\displaystyle 56 x y z \left(x + y + z - 1\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{27} = \left(\begin{array}{c}\displaystyle 4 y \left(7 x^{2} z + 7 x y z + 7 x z^{2} - 2 x z - x + 5 y z - y + 5 z^{2} - 6 z + 1\right)\\\displaystyle 56 x y z \left(x + y + z - 1\right)\\\displaystyle 112 x y z \left(- x - y - z + 1\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{28} = \left(\begin{array}{c}\displaystyle 2 x z \left(14 x y + 14 y^{2} + 14 y z - 14 y + 1\right)\\\displaystyle 4 x z \left(- 7 x y - 5 x - 7 y^{2} - 7 y z + 2 y - 5 z + 4\right)\\\displaystyle 2 x z \left(14 x y + 14 y^{2} + 14 y z - 14 y + 1\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{29} = \left(\begin{array}{c}\displaystyle 112 x y z \left(x + y + z - 1\right)\\\displaystyle 4 z \left(- 7 x^{2} y - 5 x^{2} - 7 x y^{2} - 7 x y z + 2 x y - 5 x z + 6 x + y + z - 1\right)\\\displaystyle 56 x y z \left(- x - y - z + 1\right)\end{array}\right)$$

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{30}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},0,0)\cdot\left(\begin{array}{c}0\\-1\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{30} = \left(\begin{array}{c}\displaystyle 56 x y z \left(- x - y - z + 1\right)\\\displaystyle 4 x \left(- 7 x y z - 5 x z + x - 7 y^{2} z - 7 y z^{2} + 2 y z + y - 5 z^{2} + 6 z - 1\right)\\\displaystyle 112 x y z \left(x + y + z - 1\right)\end{array}\right)$$

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{31}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},\tfrac{1}{2},0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{31} = \left(\begin{array}{c}\displaystyle 2 x y \left(- 14 x z - 14 y z - 14 z^{2} + 14 z - 1\right)\\\displaystyle 2 x y \left(- 14 x z - 14 y z - 14 z^{2} + 14 z - 1\right)\\\displaystyle 4 x y \left(7 x z + 5 x + 7 y z + 5 y + 7 z^{2} - 2 z - 4\right)\end{array}\right)$$

This DOF is associated with face 3 of the reference element.
$$\displaystyle l_{32}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{2},0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{32} = \left(\begin{array}{c}\displaystyle 112 x y z \left(- x - y - z + 1\right)\\\displaystyle 56 x y z \left(x + y + z - 1\right)\\\displaystyle 4 y \left(7 x^{2} z + 5 x^{2} + 7 x y z + 5 x y + 7 x z^{2} - 2 x z - 6 x - y - z + 1\right)\end{array}\right)$$

This DOF is associated with face 3 of the reference element.
$$\displaystyle l_{33}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},0,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{33} = \left(\begin{array}{c}\displaystyle 56 x y z \left(x + y + z - 1\right)\\\displaystyle 112 x y z \left(- x - y - z + 1\right)\\\displaystyle 4 x \left(7 x y z + 5 x y - x + 7 y^{2} z + 5 y^{2} + 7 y z^{2} - 2 y z - 6 y - z + 1\right)\end{array}\right)$$

This DOF is associated with face 3 of the reference element.
$$\displaystyle l_{34}:\boldsymbol{v}\mapsto\displaystyle\int_{R}(s_{0})\nabla\cdot\boldsymbol{v}$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1},s_{2})$$ is a parametrisation of $$R(3)$$.

$$\displaystyle \boldsymbol{\phi}_{34} = \left(\begin{array}{c}\displaystyle 5040 x y z \left(x + y + z - 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

This DOF is associated with volume 0 of the reference element.
$$\displaystyle l_{35}:\boldsymbol{v}\mapsto\displaystyle\int_{R}(s_{1})\nabla\cdot\boldsymbol{v}$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1},s_{2})$$ is a parametrisation of $$R(3)$$.

$$\displaystyle \boldsymbol{\phi}_{35} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 5040 x y z \left(x + y + z - 1\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with volume 0 of the reference element.
$$\displaystyle l_{36}:\boldsymbol{v}\mapsto\displaystyle\int_{R}(s_{2})\nabla\cdot\boldsymbol{v}$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1},s_{2})$$ is a parametrisation of $$R(3)$$.

$$\displaystyle \boldsymbol{\phi}_{36} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 5040 x y z \left(x + y + z - 1\right)\end{array}\right)$$

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

## References

• Bernardi, Christine and Raugel, Genevivève. Analysis of some finite elements for the Stokes problem, Mathematics of Computation 44, 71–79, 1985. [DOI: 10.1090/S0025-5718-1985-0771031-7] [BibTeX]

## DefElement stats

 Element added 19 April 2021 Element last updated 10 February 2022