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# Degree 2 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: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle z^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle z^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z^{2}\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{29}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(\sqrt{2})\hat{\boldsymbol{n}}_{0}$$
where $$f_{0}$$ is the 0th face;
and $$\hat{\boldsymbol{n}}_{0}$$ is the normal to facet 0.

$$\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle \sqrt{2} x \left(5 x + 5 y + 5 z - 4\right)\\\displaystyle \sqrt{2} y \left(5 x + 5 y + 5 z - 4\right)\\\displaystyle \sqrt{2} z \left(5 x + 5 y + 5 z - 4\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle x \left(- 7 x - 7 y - 4 z + 5\right)\\\displaystyle y \left(- 7 x - 7 y - 4 z + 5\right)\\\displaystyle z \left(11 x + 11 y + 14 z - 10\right)\end{array}\right)$$

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

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

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(\sqrt{6} \left(10 s_{1}^{2} - 8 s_{1} + 1\right))\hat{\boldsymbol{n}}_{0}$$
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 $$f_{0}$$.

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

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(\sqrt{2} \left(30 s_{0} s_{1} - 6 s_{0} + 15 s_{1}^{2} - 18 s_{1} + 3\right))\hat{\boldsymbol{n}}_{0}$$
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 $$f_{0}$$.

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

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(\sqrt{30} \left(6 s_{0}^{2} + 6 s_{0} s_{1} - 6 s_{0} + s_{1}^{2} - 2 s_{1} + 1\right))\hat{\boldsymbol{n}}_{0}$$
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 $$f_{0}$$.

$$\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle \sqrt{30} x \left(x - 2 y\right)\\\displaystyle \sqrt{30} y \left(- 2 x + y\right)\\\displaystyle 0\end{array}\right)$$

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

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

This DOF is associated with face 1 of the reference element.
$$\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(6 s_{1} - 2)\hat{\boldsymbol{n}}_{1}$$
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 $$f_{1}$$.

$$\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle - 7 x^{2} - 21 x z + 9 x + 6 z - 2\\\displaystyle y \left(- 7 x - 3 z + 2\right)\\\displaystyle z \left(11 x - 3 z - 1\right)\end{array}\right)$$

This DOF is associated with face 1 of the reference element.
$$\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(2 \sqrt{3} \left(2 s_{0} + s_{1} - 1\right))\hat{\boldsymbol{n}}_{1}$$
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 $$f_{1}$$.

$$\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle \sqrt{3} \left(- 7 x^{2} - 14 x y - 7 x z + 9 x + 4 y + 2 z - 2\right)\\\displaystyle \sqrt{3} y \left(5 x - 2 y - z\right)\\\displaystyle \sqrt{3} z \left(- x - 2 y - z + 1\right)\end{array}\right)$$

This DOF is associated with face 1 of the reference element.
$$\displaystyle l_{9}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(\sqrt{6} \left(10 s_{1}^{2} - 8 s_{1} + 1\right))\hat{\boldsymbol{n}}_{1}$$
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 $$f_{1}$$.

$$\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle \sqrt{6} \left(x^{2} + 8 x z - 2 x + 10 z^{2} - 8 z + 1\right)\\\displaystyle \sqrt{6} y \left(x + 4 z - 1\right)\\\displaystyle 3 \sqrt{6} z \left(- x - 2 z + 1\right)\end{array}\right)$$

This DOF is associated with face 1 of the reference element.
$$\displaystyle l_{10}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(\sqrt{2} \left(30 s_{0} s_{1} - 6 s_{0} + 15 s_{1}^{2} - 18 s_{1} + 3\right))\hat{\boldsymbol{n}}_{1}$$
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 $$f_{1}$$.

$$\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 3 \sqrt{2} \left(x^{2} + 2 x y + 6 x z - 2 x + 10 y z - 2 y + 5 z^{2} - 6 z + 1\right)\\\displaystyle 3 \sqrt{2} y \left(y - 2 z\right)\\\displaystyle 6 \sqrt{2} z \left(- x - 2 y - z + 1\right)\end{array}\right)$$

This DOF is associated with face 1 of the reference element.
$$\displaystyle l_{11}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(\sqrt{30} \left(6 s_{0}^{2} + 6 s_{0} s_{1} - 6 s_{0} + s_{1}^{2} - 2 s_{1} + 1\right))\hat{\boldsymbol{n}}_{1}$$
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 $$f_{1}$$.

$$\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle \sqrt{30} \left(x^{2} + 6 x y + 2 x z - 2 x + 6 y^{2} + 6 y z - 6 y + z^{2} - 2 z + 1\right)\\\displaystyle \sqrt{30} y \left(- 2 x - 3 y - 2 z + 2\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 1 of the reference element.
$$\displaystyle l_{12}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(\sqrt{2})\hat{\boldsymbol{n}}_{2}$$
where $$f_{2}$$ is the 2nd face;
and $$\hat{\boldsymbol{n}}_{2}$$ is the normal to facet 2.

$$\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle \sqrt{2} x \left(1 - 5 y\right)\\\displaystyle \sqrt{2} \left(- 5 y^{2} + 6 y - 1\right)\\\displaystyle \sqrt{2} z \left(1 - 5 y\right)\end{array}\right)$$

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(6 s_{1} - 2)\hat{\boldsymbol{n}}_{2}$$
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 $$f_{2}$$.

$$\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle x \left(7 y + 3 z - 2\right)\\\displaystyle 7 y^{2} + 21 y z - 9 y - 6 z + 2\\\displaystyle z \left(- 11 y + 3 z + 1\right)\end{array}\right)$$

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{14}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(2 \sqrt{3} \left(2 s_{0} + s_{1} - 1\right))\hat{\boldsymbol{n}}_{2}$$
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 $$f_{2}$$.

$$\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle \sqrt{3} x \left(2 x - 5 y + z\right)\\\displaystyle \sqrt{3} \left(14 x y - 4 x + 7 y^{2} + 7 y z - 9 y - 2 z + 2\right)\\\displaystyle \sqrt{3} z \left(2 x + y + z - 1\right)\end{array}\right)$$

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{15}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(\sqrt{6} \left(10 s_{1}^{2} - 8 s_{1} + 1\right))\hat{\boldsymbol{n}}_{2}$$
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 $$f_{2}$$.

$$\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle \sqrt{6} x \left(- y - 4 z + 1\right)\\\displaystyle \sqrt{6} \left(- y^{2} - 8 y z + 2 y - 10 z^{2} + 8 z - 1\right)\\\displaystyle 3 \sqrt{6} z \left(y + 2 z - 1\right)\end{array}\right)$$

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{16}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(\sqrt{2} \left(30 s_{0} s_{1} - 6 s_{0} + 15 s_{1}^{2} - 18 s_{1} + 3\right))\hat{\boldsymbol{n}}_{2}$$
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 $$f_{2}$$.

$$\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle 3 \sqrt{2} x \left(- x + 2 z\right)\\\displaystyle 3 \sqrt{2} \left(- 2 x y - 10 x z + 2 x - y^{2} - 6 y z + 2 y - 5 z^{2} + 6 z - 1\right)\\\displaystyle 6 \sqrt{2} z \left(2 x + y + z - 1\right)\end{array}\right)$$

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{17}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(\sqrt{30} \left(6 s_{0}^{2} + 6 s_{0} s_{1} - 6 s_{0} + s_{1}^{2} - 2 s_{1} + 1\right))\hat{\boldsymbol{n}}_{2}$$
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 $$f_{2}$$.

$$\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle \sqrt{30} x \left(3 x + 2 y + 2 z - 2\right)\\\displaystyle \sqrt{30} \left(- 6 x^{2} - 6 x y - 6 x z + 6 x - y^{2} - 2 y z + 2 y - z^{2} + 2 z - 1\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{18}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(\sqrt{2})\hat{\boldsymbol{n}}_{3}$$
where $$f_{3}$$ is the 3rd face;
and $$\hat{\boldsymbol{n}}_{3}$$ is the normal to facet 3.

$$\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle \sqrt{2} x \left(5 z - 1\right)\\\displaystyle \sqrt{2} y \left(5 z - 1\right)\\\displaystyle \sqrt{2} \left(5 z^{2} - 6 z + 1\right)\end{array}\right)$$

This DOF is associated with face 3 of the reference element.
$$\displaystyle l_{19}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(6 s_{1} - 2)\hat{\boldsymbol{n}}_{3}$$
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 $$f_{3}$$.

$$\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle x \left(- 3 y - 7 z + 2\right)\\\displaystyle y \left(- 3 y + 11 z - 1\right)\\\displaystyle - 21 y z + 6 y - 7 z^{2} + 9 z - 2\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{20} = \left(\begin{array}{c}\displaystyle \sqrt{3} x \left(- 2 x - y + 5 z\right)\\\displaystyle \sqrt{3} y \left(- 2 x - y - z + 1\right)\\\displaystyle \sqrt{3} \left(- 14 x z + 4 x - 7 y z + 2 y - 7 z^{2} + 9 z - 2\right)\end{array}\right)$$

This DOF is associated with face 3 of the reference element.
$$\displaystyle l_{21}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(\sqrt{6} \left(10 s_{1}^{2} - 8 s_{1} + 1\right))\hat{\boldsymbol{n}}_{3}$$
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 $$f_{3}$$.

$$\displaystyle \boldsymbol{\phi}_{21} = \left(\begin{array}{c}\displaystyle \sqrt{6} x \left(4 y + z - 1\right)\\\displaystyle 3 \sqrt{6} y \left(- 2 y - z + 1\right)\\\displaystyle \sqrt{6} \left(10 y^{2} + 8 y z - 8 y + z^{2} - 2 z + 1\right)\end{array}\right)$$

This DOF is associated with face 3 of the reference element.
$$\displaystyle l_{22}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(\sqrt{2} \left(30 s_{0} s_{1} - 6 s_{0} + 15 s_{1}^{2} - 18 s_{1} + 3\right))\hat{\boldsymbol{n}}_{3}$$
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 $$f_{3}$$.

$$\displaystyle \boldsymbol{\phi}_{22} = \left(\begin{array}{c}\displaystyle 3 \sqrt{2} x \left(x - 2 y\right)\\\displaystyle 6 \sqrt{2} y \left(- 2 x - y - z + 1\right)\\\displaystyle 3 \sqrt{2} \left(10 x y + 2 x z - 2 x + 5 y^{2} + 6 y z - 6 y + z^{2} - 2 z + 1\right)\end{array}\right)$$

This DOF is associated with face 3 of the reference element.
$$\displaystyle l_{23}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(\sqrt{30} \left(6 s_{0}^{2} + 6 s_{0} s_{1} - 6 s_{0} + s_{1}^{2} - 2 s_{1} + 1\right))\hat{\boldsymbol{n}}_{3}$$
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 $$f_{3}$$.

$$\displaystyle \boldsymbol{\phi}_{23} = \left(\begin{array}{c}\displaystyle \sqrt{30} x \left(- 3 x - 2 y - 2 z + 2\right)\\\displaystyle 0\\\displaystyle \sqrt{30} \left(6 x^{2} + 6 x y + 6 x z - 6 x + y^{2} + 2 y z - 2 y + z^{2} - 2 z + 1\right)\end{array}\right)$$

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

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

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

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

This DOF is associated with volume 0 of the reference element.
$$\displaystyle l_{26}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}- s_{1}\\s_{0}\\0\end{array}\right)$$
where $$R$$ is the reference element;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$R$$.

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

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

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

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

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

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

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

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