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# Degree 1 Brezzi–Douglas–Marini on a tetrahedron

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In this example:
• $$R$$ 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:
$$\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(- s_{0} - s_{1} + 1)\hat{\boldsymbol{n}}_{0}$$
where $$f_{0}$$ is the 0th face;
$$\hat{\boldsymbol{n}}_{0}$$ is the normal to facet 0;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$f_{0}$$.

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

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{0}$$
where $$f_{0}$$ is the 0th face;
$$\hat{\boldsymbol{n}}_{0}$$ is the normal to facet 0;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$f_{0}$$.

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

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(s_{1})\hat{\boldsymbol{n}}_{0}$$
where $$f_{0}$$ is the 0th face;
$$\hat{\boldsymbol{n}}_{0}$$ is the normal to facet 0;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$f_{0}$$.

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

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(- s_{0} - s_{1} + 1)\hat{\boldsymbol{n}}_{1}$$
where $$f_{1}$$ is the 1st face;
$$\hat{\boldsymbol{n}}_{1}$$ is the normal to facet 1;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$f_{1}$$.

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

This DOF is associated with face 1 of the reference element.
$$\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{1}$$
where $$f_{1}$$ is the 1st face;
$$\hat{\boldsymbol{n}}_{1}$$ is the normal to facet 1;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$f_{1}$$.

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

This DOF is associated with face 1 of the reference element.
$$\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(s_{1})\hat{\boldsymbol{n}}_{1}$$
where $$f_{1}$$ is the 1st face;
$$\hat{\boldsymbol{n}}_{1}$$ is the normal to facet 1;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$f_{1}$$.

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

This DOF is associated with face 1 of the reference element.
$$\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(- s_{0} - s_{1} + 1)\hat{\boldsymbol{n}}_{2}$$
where $$f_{2}$$ is the 2nd face;
$$\hat{\boldsymbol{n}}_{2}$$ is the normal to facet 2;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$f_{2}$$.

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

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{2}$$
where $$f_{2}$$ is the 2nd face;
$$\hat{\boldsymbol{n}}_{2}$$ is the normal to facet 2;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$f_{2}$$.

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

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(s_{1})\hat{\boldsymbol{n}}_{2}$$
where $$f_{2}$$ is the 2nd face;
$$\hat{\boldsymbol{n}}_{2}$$ is the normal to facet 2;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$f_{2}$$.

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

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{9}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(- s_{0} - s_{1} + 1)\hat{\boldsymbol{n}}_{3}$$
where $$f_{3}$$ is the 3rd face;
$$\hat{\boldsymbol{n}}_{3}$$ is the normal to facet 3;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$f_{3}$$.

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

This DOF is associated with face 3 of the reference element.
$$\displaystyle l_{10}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{3}$$
where $$f_{3}$$ is the 3rd face;
$$\hat{\boldsymbol{n}}_{3}$$ is the normal to facet 3;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$f_{3}$$.

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

This DOF is associated with face 3 of the reference element.
$$\displaystyle l_{11}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(s_{1})\hat{\boldsymbol{n}}_{3}$$
where $$f_{3}$$ is the 3rd face;
$$\hat{\boldsymbol{n}}_{3}$$ is the normal to facet 3;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$f_{3}$$.

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

This DOF is associated with face 3 of the reference element.