an encyclopedia of finite element definitions

# Degree 2 Q H(div) on a hexahedron

◀ Back to Q H(div) definition page In this example:
• $$R$$ is the reference hexahedron. The following numbering of the subentities of the reference is used:
• • $$\mathcal{V}$$ is spanned by: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2} z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2} y\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2} y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2} z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y^{2} z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y z^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x z^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y z^{2}\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{35}\}$$
• Functionals and basis functions: $$\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(s_{0} s_{1} - s_{0} - s_{1} + 1)\hat{\boldsymbol{n}}_{0}$$
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 $$f_{0}$$.

$$\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 108 x y z^{2} - 144 x y z + 36 x y - 72 x z^{2} + 96 x z - 24 x - 72 y z^{2} + 96 y z - 24 y + 48 z^{2} - 64 z + 16\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - 108 x y z^{2} + 144 x y z - 36 x y + 72 x z^{2} - 96 x z + 24 x + 36 y z^{2} - 48 y z + 12 y - 24 z^{2} + 32 z - 8\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - 108 x y z^{2} + 144 x y z - 36 x y + 36 x z^{2} - 48 x z + 12 x + 72 y z^{2} - 96 y z + 24 y - 24 z^{2} + 32 z - 8\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 108 x y z^{2} - 144 x y z + 36 x y - 36 x z^{2} + 48 x z - 12 x - 36 y z^{2} + 48 y z - 12 y + 12 z^{2} - 16 z + 4\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - 108 x y^{2} z + 72 x y^{2} + 144 x y z - 96 x y - 36 x z + 24 x + 72 y^{2} z - 48 y^{2} - 96 y z + 64 y + 24 z - 16\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 108 x y^{2} z - 72 x y^{2} - 144 x y z + 96 x y + 36 x z - 24 x - 36 y^{2} z + 24 y^{2} + 48 y z - 32 y - 12 z + 8\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 108 x y^{2} z - 36 x y^{2} - 144 x y z + 48 x y + 36 x z - 12 x - 72 y^{2} z + 24 y^{2} + 96 y z - 32 y - 24 z + 8\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - 108 x y^{2} z + 36 x y^{2} + 144 x y z - 48 x y - 36 x z + 12 x + 36 y^{2} z - 12 y^{2} - 48 y z + 16 y + 12 z - 4\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 108 x^{2} y z - 72 x^{2} y - 72 x^{2} z + 48 x^{2} - 144 x y z + 96 x y + 96 x z - 64 x + 36 y z - 24 y - 24 z + 16\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle - 108 x^{2} y z + 72 x^{2} y + 36 x^{2} z - 24 x^{2} + 144 x y z - 96 x y - 48 x z + 32 x - 36 y z + 24 y + 12 z - 8\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle - 108 x^{2} y z + 36 x^{2} y + 72 x^{2} z - 24 x^{2} + 144 x y z - 48 x y - 96 x z + 32 x - 36 y z + 12 y + 24 z - 8\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 108 x^{2} y z - 36 x^{2} y - 36 x^{2} z + 12 x^{2} - 144 x y z + 48 x y + 48 x z - 16 x + 36 y z - 12 y - 12 z + 4\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 4 x \left(27 x y z - 18 x y - 18 x z + 12 x - 18 y z + 12 y + 12 z - 8\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 3 of the reference element. $$\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(s_{0} \cdot \left(1 - s_{1}\right))\hat{\boldsymbol{n}}_{3}$$
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 $$f_{3}$$.

$$\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 4 x \left(- 27 x y z + 18 x y + 9 x z - 6 x + 18 y z - 12 y - 6 z + 4\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 4 x \left(- 27 x y z + 9 x y + 18 x z - 6 x + 18 y z - 6 y - 12 z + 4\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 3 of the reference element. $$\displaystyle l_{15}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(s_{0} s_{1})\hat{\boldsymbol{n}}_{3}$$
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 $$f_{3}$$.

$$\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 4 x \left(27 x y z - 9 x y - 9 x z + 3 x - 18 y z + 6 y + 6 z - 2\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 3 of the reference element. $$\displaystyle l_{16}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot(s_{0} s_{1} - s_{0} - s_{1} + 1)\hat{\boldsymbol{n}}_{4}$$
where $$f_{4}$$ is the 4th face;
$$\hat{\boldsymbol{n}}_{4}$$ is the normal to facet 4;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$f_{4}$$.

$$\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 y \left(- 27 x y z + 18 x y + 18 x z - 12 x + 18 y z - 12 y - 12 z + 8\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 4 of the reference element. $$\displaystyle l_{17}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot(s_{0} \cdot \left(1 - s_{1}\right))\hat{\boldsymbol{n}}_{4}$$
where $$f_{4}$$ is the 4th face;
$$\hat{\boldsymbol{n}}_{4}$$ is the normal to facet 4;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$f_{4}$$.

$$\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 y \left(27 x y z - 18 x y - 18 x z + 12 x - 9 y z + 6 y + 6 z - 4\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 4 of the reference element. $$\displaystyle l_{18}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot(s_{1} \cdot \left(1 - s_{0}\right))\hat{\boldsymbol{n}}_{4}$$
where $$f_{4}$$ is the 4th face;
$$\hat{\boldsymbol{n}}_{4}$$ is the normal to facet 4;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$f_{4}$$.

$$\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 y \left(27 x y z - 9 x y - 18 x z + 6 x - 18 y z + 6 y + 12 z - 4\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 4 of the reference element. $$\displaystyle l_{19}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot(s_{0} s_{1})\hat{\boldsymbol{n}}_{4}$$
where $$f_{4}$$ is the 4th face;
$$\hat{\boldsymbol{n}}_{4}$$ is the normal to facet 4;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$f_{4}$$.

$$\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 y \left(- 27 x y z + 9 x y + 18 x z - 6 x + 9 y z - 3 y - 6 z + 2\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 4 of the reference element. $$\displaystyle l_{20}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot(s_{0} s_{1} - s_{0} - s_{1} + 1)\hat{\boldsymbol{n}}_{5}$$
where $$f_{5}$$ is the 5th face;
$$\hat{\boldsymbol{n}}_{5}$$ is the normal to facet 5;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$f_{5}$$.

$$\displaystyle \boldsymbol{\phi}_{20} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 z \left(27 x y z - 18 x y - 18 x z + 12 x - 18 y z + 12 y + 12 z - 8\right)\end{array}\right)$$

This DOF is associated with face 5 of the reference element. $$\displaystyle l_{21}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot(s_{0} \cdot \left(1 - s_{1}\right))\hat{\boldsymbol{n}}_{5}$$
where $$f_{5}$$ is the 5th face;
$$\hat{\boldsymbol{n}}_{5}$$ is the normal to facet 5;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$f_{5}$$.

$$\displaystyle \boldsymbol{\phi}_{21} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 z \left(- 27 x y z + 18 x y + 18 x z - 12 x + 9 y z - 6 y - 6 z + 4\right)\end{array}\right)$$

This DOF is associated with face 5 of the reference element. $$\displaystyle l_{22}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot(s_{1} \cdot \left(1 - s_{0}\right))\hat{\boldsymbol{n}}_{5}$$
where $$f_{5}$$ is the 5th face;
$$\hat{\boldsymbol{n}}_{5}$$ is the normal to facet 5;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$f_{5}$$.

$$\displaystyle \boldsymbol{\phi}_{22} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 z \left(- 27 x y z + 18 x y + 9 x z - 6 x + 18 y z - 12 y - 6 z + 4\right)\end{array}\right)$$

This DOF is associated with face 5 of the reference element. $$\displaystyle l_{23}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot(s_{0} s_{1})\hat{\boldsymbol{n}}_{5}$$
where $$f_{5}$$ is the 5th face;
$$\hat{\boldsymbol{n}}_{5}$$ is the normal to facet 5;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$f_{5}$$.

$$\displaystyle \boldsymbol{\phi}_{23} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 z \left(27 x y z - 18 x y - 9 x z + 6 x - 9 y z + 6 y + 3 z - 2\right)\end{array}\right)$$

This DOF is associated with face 5 of the reference element. $$\displaystyle l_{24}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}s_{1} s_{2} - s_{1} - s_{2} + 1\\0\\0\end{array}\right)$$
where $$R$$ is the reference element;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$R$$.

$$\displaystyle \boldsymbol{\phi}_{24} = \left(\begin{array}{c}\displaystyle 24 x \left(- 9 x y z + 6 x y + 6 x z - 4 x + 9 y z - 6 y - 6 z + 4\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{25} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 24 y \left(- 9 x y z + 6 x y + 9 x z - 6 x + 6 y z - 4 y - 6 z + 4\right)\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{26} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 24 z \left(- 9 x y z + 9 x y + 6 x z - 6 x + 6 y z - 6 y - 4 z + 4\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{27} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 24 y \left(9 x y z - 6 x y - 9 x z + 6 x - 3 y z + 2 y + 3 z - 2\right)\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{28} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 24 z \left(9 x y z - 9 x y - 6 x z + 6 x - 3 y z + 3 y + 2 z - 2\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{29} = \left(\begin{array}{c}\displaystyle 24 x \left(9 x y z - 6 x y - 3 x z + 2 x - 9 y z + 6 y + 3 z - 2\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

This DOF is associated with volume 0 of the reference element. $$\displaystyle l_{30}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\0\\s_{1} \cdot \left(1 - s_{0}\right)\end{array}\right)$$
where $$R$$ is the reference element;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$R$$.

$$\displaystyle \boldsymbol{\phi}_{30} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 24 z \left(9 x y z - 9 x y - 3 x z + 3 x - 6 y z + 6 y + 2 z - 2\right)\end{array}\right)$$

This DOF is associated with volume 0 of the reference element. $$\displaystyle l_{31}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\0\\s_{0} s_{1}\end{array}\right)$$
where $$R$$ is the reference element;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$R$$.

$$\displaystyle \boldsymbol{\phi}_{31} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 24 z \left(- 9 x y z + 9 x y + 3 x z - 3 x + 3 y z - 3 y - z + 1\right)\end{array}\right)$$

This DOF is associated with volume 0 of the reference element. $$\displaystyle l_{32}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}s_{2} \cdot \left(1 - s_{1}\right)\\0\\0\end{array}\right)$$
where $$R$$ is the reference element;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$R$$.

$$\displaystyle \boldsymbol{\phi}_{32} = \left(\begin{array}{c}\displaystyle 24 x \left(9 x y z - 3 x y - 6 x z + 2 x - 9 y z + 3 y + 6 z - 2\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

This DOF is associated with volume 0 of the reference element. $$\displaystyle l_{33}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\s_{2} \cdot \left(1 - s_{0}\right)\\0\end{array}\right)$$
where $$R$$ is the reference element;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$R$$.

$$\displaystyle \boldsymbol{\phi}_{33} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 24 y \left(9 x y z - 3 x y - 9 x z + 3 x - 6 y z + 2 y + 6 z - 2\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with volume 0 of the reference element. $$\displaystyle l_{34}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\s_{0} s_{2}\\0\end{array}\right)$$
where $$R$$ is the reference element;
and $$s_{0},s_{1},s_{2}$$ is a parametrisation of $$R$$.

$$\displaystyle \boldsymbol{\phi}_{34} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 24 y \left(- 9 x y z + 3 x y + 9 x z - 3 x + 3 y z - y - 3 z + 1\right)\\\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{35} = \left(\begin{array}{c}\displaystyle 24 x \left(- 9 x y z + 3 x y + 3 x z - x + 9 y z - 3 y - 3 z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

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