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# Degree 1 serendipity H(div) on a hexahedron

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

This DOF is associated with face 4 of the reference element.
$$\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{4}$$
where $$f_{4}$$ is the 4th face;
and $$\hat{\boldsymbol{n}}_{4}$$ is the normal to facet 4.

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

This DOF is associated with face 4 of the reference element.
$$\displaystyle l_{14}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot(s_{1})\hat{\boldsymbol{n}}_{4}$$
where $$f_{4}$$ is the 4th face;
and $$\hat{\boldsymbol{n}}_{4}$$ is the normal to facet 4.

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

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

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

This DOF is associated with face 5 of the reference element.
$$\displaystyle l_{16}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{5}$$
where $$f_{5}$$ is the 5th face;
and $$\hat{\boldsymbol{n}}_{5}$$ is the normal to facet 5.

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

This DOF is associated with face 5 of the reference element.
$$\displaystyle l_{17}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot(s_{1})\hat{\boldsymbol{n}}_{5}$$
where $$f_{5}$$ is the 5th face;
and $$\hat{\boldsymbol{n}}_{5}$$ is the normal to facet 5.

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

This DOF is associated with face 5 of the reference element.