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# Degree 2 Nédélec (first kind) on a hexahedron

◀ Back to Nédélec (first kind) 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: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle z^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y z^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y^{2} z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y^{2} z^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x z^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y z^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y^{2} z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y^{2} z^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle z^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y z^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x z^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y z^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2} z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2} z^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2} y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2} y z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2} y z^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y^{2} z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y^{2} z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x^{2} z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x^{2} y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x^{2} y z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x^{2} y^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x^{2} y^{2} z\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{53}\}$$
• Functionals and basis functions: $$\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{0}$$
where $$e_{0}$$ is the 0th edge;
and $$\hat{\boldsymbol{t}}_{0}$$ is the tangent to edge 0.

$$\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle - 54 x y^{2} z^{2} + 72 x y^{2} z - 18 x y^{2} + 72 x y z^{2} - 96 x y z + 24 x y - 18 x z^{2} + 24 x z - 6 x + 36 y^{2} z^{2} - 48 y^{2} z + 12 y^{2} - 48 y z^{2} + 64 y z - 16 y + 12 z^{2} - 16 z + 4\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 0 of the reference element. $$\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{0}$$
where $$e_{0}$$ is the 0th edge;
and $$\hat{\boldsymbol{t}}_{0}$$ is the tangent to edge 0.

$$\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 54 x y^{2} z^{2} - 72 x y^{2} z + 18 x y^{2} - 72 x y z^{2} + 96 x y z - 24 x y + 18 x z^{2} - 24 x z + 6 x - 18 y^{2} z^{2} + 24 y^{2} z - 6 y^{2} + 24 y z^{2} - 32 y z + 8 y - 6 z^{2} + 8 z - 2\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 0 of the reference element. $$\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{1}$$
where $$e_{1}$$ is the 1st edge;
and $$\hat{\boldsymbol{t}}_{1}$$ is the tangent to edge 1.

$$\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - 54 x^{2} y z^{2} + 72 x^{2} y z - 18 x^{2} y + 36 x^{2} z^{2} - 48 x^{2} z + 12 x^{2} + 72 x y z^{2} - 96 x y z + 24 x y - 48 x z^{2} + 64 x z - 16 x - 18 y z^{2} + 24 y z - 6 y + 12 z^{2} - 16 z + 4\\\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 1 of the reference element. $$\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{1}$$
where $$e_{1}$$ is the 1st edge;
and $$\hat{\boldsymbol{t}}_{1}$$ is the tangent to edge 1.

$$\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 54 x^{2} y z^{2} - 72 x^{2} y z + 18 x^{2} y - 18 x^{2} z^{2} + 24 x^{2} z - 6 x^{2} - 72 x y z^{2} + 96 x y z - 24 x y + 24 x z^{2} - 32 x z + 8 x + 18 y z^{2} - 24 y z + 6 y - 6 z^{2} + 8 z - 2\\\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 1 of the reference element. $$\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{2}$$
where $$e_{2}$$ is the 2nd edge;
and $$\hat{\boldsymbol{t}}_{2}$$ is the tangent to edge 2.

$$\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - 54 x^{2} y^{2} z + 36 x^{2} y^{2} + 72 x^{2} y z - 48 x^{2} y - 18 x^{2} z + 12 x^{2} + 72 x y^{2} z - 48 x y^{2} - 96 x y z + 64 x y + 24 x z - 16 x - 18 y^{2} z + 12 y^{2} + 24 y z - 16 y - 6 z + 4\end{array}\right)$$

This DOF is associated with edge 2 of the reference element. $$\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{2}$$
where $$e_{2}$$ is the 2nd edge;
and $$\hat{\boldsymbol{t}}_{2}$$ is the tangent to edge 2.

$$\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 54 x^{2} y^{2} z - 18 x^{2} y^{2} - 72 x^{2} y z + 24 x^{2} y + 18 x^{2} z - 6 x^{2} - 72 x y^{2} z + 24 x y^{2} + 96 x y z - 32 x y - 24 x z + 8 x + 18 y^{2} z - 6 y^{2} - 24 y z + 8 y + 6 z - 2\end{array}\right)$$

This DOF is associated with edge 2 of the reference element. $$\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{3}$$
where $$e_{3}$$ is the 3th edge;
and $$\hat{\boldsymbol{t}}_{3}$$ is the tangent to edge 3.

$$\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 x \left(- 27 x y z^{2} + 36 x y z - 9 x y + 18 x z^{2} - 24 x z + 6 x + 18 y z^{2} - 24 y z + 6 y - 12 z^{2} + 16 z - 4\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 3 of the reference element. $$\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{3}$$
where $$e_{3}$$ is the 3th edge;
and $$\hat{\boldsymbol{t}}_{3}$$ is the tangent to edge 3.

$$\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 x \left(27 x y z^{2} - 36 x y z + 9 x y - 9 x z^{2} + 12 x z - 3 x - 18 y z^{2} + 24 y z - 6 y + 6 z^{2} - 8 z + 2\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 3 of the reference element. $$\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{4}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{4}$$
where $$e_{4}$$ is the 4th edge;
and $$\hat{\boldsymbol{t}}_{4}$$ is the tangent to edge 4.

$$\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 2 x \left(- 27 x y^{2} z + 18 x y^{2} + 36 x y z - 24 x y - 9 x z + 6 x + 18 y^{2} z - 12 y^{2} - 24 y z + 16 y + 6 z - 4\right)\end{array}\right)$$

This DOF is associated with edge 4 of the reference element. $$\displaystyle l_{9}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{4}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{4}$$
where $$e_{4}$$ is the 4th edge;
and $$\hat{\boldsymbol{t}}_{4}$$ is the tangent to edge 4.

$$\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 2 x \left(27 x y^{2} z - 9 x y^{2} - 36 x y z + 12 x y + 9 x z - 3 x - 18 y^{2} z + 6 y^{2} + 24 y z - 8 y - 6 z + 2\right)\end{array}\right)$$

This DOF is associated with edge 4 of the reference element. $$\displaystyle l_{10}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{5}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{5}$$
where $$e_{5}$$ is the 5th edge;
and $$\hat{\boldsymbol{t}}_{5}$$ is the tangent to edge 5.

$$\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 2 y \left(- 27 x y z^{2} + 36 x y z - 9 x y + 18 x z^{2} - 24 x z + 6 x + 18 y z^{2} - 24 y z + 6 y - 12 z^{2} + 16 z - 4\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 5 of the reference element. $$\displaystyle l_{11}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{5}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{5}$$
where $$e_{5}$$ is the 5th edge;
and $$\hat{\boldsymbol{t}}_{5}$$ is the tangent to edge 5.

$$\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 2 y \left(27 x y z^{2} - 36 x y z + 9 x y - 18 x z^{2} + 24 x z - 6 x - 9 y z^{2} + 12 y z - 3 y + 6 z^{2} - 8 z + 2\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 5 of the reference element. $$\displaystyle l_{12}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{6}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{6}$$
where $$e_{6}$$ is the 6th edge;
and $$\hat{\boldsymbol{t}}_{6}$$ is the tangent to edge 6.

$$\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 2 y \left(- 27 x^{2} y z + 18 x^{2} y + 18 x^{2} z - 12 x^{2} + 36 x y z - 24 x y - 24 x z + 16 x - 9 y z + 6 y + 6 z - 4\right)\end{array}\right)$$

This DOF is associated with edge 6 of the reference element. $$\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{6}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{6}$$
where $$e_{6}$$ is the 6th edge;
and $$\hat{\boldsymbol{t}}_{6}$$ is the tangent to edge 6.

$$\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 2 y \left(27 x^{2} y z - 9 x^{2} y - 18 x^{2} z + 6 x^{2} - 36 x y z + 12 x y + 24 x z - 8 x + 9 y z - 3 y - 6 z + 2\right)\end{array}\right)$$

This DOF is associated with edge 6 of the reference element. $$\displaystyle l_{14}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{7}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{7}$$
where $$e_{7}$$ is the 7th edge;
and $$\hat{\boldsymbol{t}}_{7}$$ is the tangent to edge 7.

$$\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 2 x y \left(- 27 x y z + 18 x y + 18 x z - 12 x + 18 y z - 12 y - 12 z + 8\right)\end{array}\right)$$

This DOF is associated with edge 7 of the reference element. $$\displaystyle l_{15}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{7}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{7}$$
where $$e_{7}$$ is the 7th edge;
and $$\hat{\boldsymbol{t}}_{7}$$ is the tangent to edge 7.

$$\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 2 x y \left(27 x y z - 9 x y - 18 x z + 6 x - 18 y z + 6 y + 12 z - 4\right)\end{array}\right)$$

This DOF is associated with edge 7 of the reference element. $$\displaystyle l_{16}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{8}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{8}$$
where $$e_{8}$$ is the 8th edge;
and $$\hat{\boldsymbol{t}}_{8}$$ is the tangent to edge 8.

$$\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle 2 z \left(- 27 x y^{2} z + 18 x y^{2} + 36 x y z - 24 x y - 9 x z + 6 x + 18 y^{2} z - 12 y^{2} - 24 y z + 16 y + 6 z - 4\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 8 of the reference element. $$\displaystyle l_{17}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{8}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{8}$$
where $$e_{8}$$ is the 8th edge;
and $$\hat{\boldsymbol{t}}_{8}$$ is the tangent to edge 8.

$$\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle 2 z \left(27 x y^{2} z - 18 x y^{2} - 36 x y z + 24 x y + 9 x z - 6 x - 9 y^{2} z + 6 y^{2} + 12 y z - 8 y - 3 z + 2\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 8 of the reference element. $$\displaystyle l_{18}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{9}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{9}$$
where $$e_{9}$$ is the 9th edge;
and $$\hat{\boldsymbol{t}}_{9}$$ is the tangent to edge 9.

$$\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 z \left(- 27 x^{2} y z + 18 x^{2} y + 18 x^{2} z - 12 x^{2} + 36 x y z - 24 x y - 24 x z + 16 x - 9 y z + 6 y + 6 z - 4\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 9 of the reference element. $$\displaystyle l_{19}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{9}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{9}$$
where $$e_{9}$$ is the 9th edge;
and $$\hat{\boldsymbol{t}}_{9}$$ is the tangent to edge 9.

$$\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 z \left(27 x^{2} y z - 18 x^{2} y - 9 x^{2} z + 6 x^{2} - 36 x y z + 24 x y + 12 x z - 8 x + 9 y z - 6 y - 3 z + 2\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 9 of the reference element. $$\displaystyle l_{20}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{10}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{10}$$
where $$e_{10}$$ is the 10th edge;
and $$\hat{\boldsymbol{t}}_{10}$$ is the tangent to edge 10.

$$\displaystyle \boldsymbol{\phi}_{20} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 x z \left(- 27 x y z + 18 x y + 18 x z - 12 x + 18 y z - 12 y - 12 z + 8\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 10 of the reference element. $$\displaystyle l_{21}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{10}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{10}$$
where $$e_{10}$$ is the 10th edge;
and $$\hat{\boldsymbol{t}}_{10}$$ is the tangent to edge 10.

$$\displaystyle \boldsymbol{\phi}_{21} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 x z \left(27 x y z - 18 x y - 9 x z + 6 x - 18 y z + 12 y + 6 z - 4\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 10 of the reference element. $$\displaystyle l_{22}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{11}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{11}$$
where $$e_{11}$$ is the 11th edge;
and $$\hat{\boldsymbol{t}}_{11}$$ is the tangent to edge 11.

$$\displaystyle \boldsymbol{\phi}_{22} = \left(\begin{array}{c}\displaystyle 2 y z \left(- 27 x y z + 18 x y + 18 x z - 12 x + 18 y z - 12 y - 12 z + 8\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 11 of the reference element. $$\displaystyle l_{23}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{11}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{11}$$
where $$e_{11}$$ is the 11th edge;
and $$\hat{\boldsymbol{t}}_{11}$$ is the tangent to edge 11.

$$\displaystyle \boldsymbol{\phi}_{23} = \left(\begin{array}{c}\displaystyle 2 y z \left(27 x y z - 18 x y - 18 x z + 12 x - 9 y z + 6 y + 6 z - 4\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 11 of the reference element. $$\displaystyle l_{24}:v\mapsto\displaystyle\int_{f_{0}}(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1 - s_{1}\\\displaystyle 0\end{array}\right))v$$
where $$f_{0}$$ is the 0th face.

$$\displaystyle \boldsymbol{\phi}_{24} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 12 x \left(9 x y z^{2} - 12 x y z + 3 x y - 6 x z^{2} + 8 x z - 2 x - 9 y z^{2} + 12 y z - 3 y + 6 z^{2} - 8 z + 2\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 0 of the reference element. $$\displaystyle l_{25}:v\mapsto\displaystyle\int_{f_{0}}(\left(\begin{array}{c}\displaystyle s_{0} - 1\\\displaystyle 0\\\displaystyle 0\end{array}\right))v$$
where $$f_{0}$$ is the 0th face.

$$\displaystyle \boldsymbol{\phi}_{25} = \left(\begin{array}{c}\displaystyle 12 y \left(- 9 x y z^{2} + 12 x y z - 3 x y + 9 x z^{2} - 12 x z + 3 x + 6 y z^{2} - 8 y z + 2 y - 6 z^{2} + 8 z - 2\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 0 of the reference element. $$\displaystyle l_{26}:v\mapsto\displaystyle\int_{f_{0}}(\left(\begin{array}{c}\displaystyle - s_{0}\\\displaystyle 0\\\displaystyle 0\end{array}\right))v$$
where $$f_{0}$$ is the 0th face.

$$\displaystyle \boldsymbol{\phi}_{26} = \left(\begin{array}{c}\displaystyle 12 y \left(9 x y z^{2} - 12 x y z + 3 x y - 9 x z^{2} + 12 x z - 3 x - 3 y z^{2} + 4 y z - y + 3 z^{2} - 4 z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 0 of the reference element. $$\displaystyle l_{27}:v\mapsto\displaystyle\int_{f_{0}}(\left(\begin{array}{c}\displaystyle 0\\\displaystyle s_{1}\\\displaystyle 0\end{array}\right))v$$
where $$f_{0}$$ is the 0th face.

$$\displaystyle \boldsymbol{\phi}_{27} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 12 x \left(- 9 x y z^{2} + 12 x y z - 3 x y + 3 x z^{2} - 4 x z + x + 9 y z^{2} - 12 y z + 3 y - 3 z^{2} + 4 z - 1\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 0 of the reference element. $$\displaystyle l_{28}:v\mapsto\displaystyle\int_{f_{1}}(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1 - s_{1}\end{array}\right))v$$
where $$f_{1}$$ is the 1st face.

$$\displaystyle \boldsymbol{\phi}_{28} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 12 x \left(9 x y^{2} z - 6 x y^{2} - 12 x y z + 8 x y + 3 x z - 2 x - 9 y^{2} z + 6 y^{2} + 12 y z - 8 y - 3 z + 2\right)\end{array}\right)$$

This DOF is associated with face 1 of the reference element. $$\displaystyle l_{29}:v\mapsto\displaystyle\int_{f_{1}}(\left(\begin{array}{c}\displaystyle s_{0} - 1\\\displaystyle 0\\\displaystyle 0\end{array}\right))v$$
where $$f_{1}$$ is the 1st face.

$$\displaystyle \boldsymbol{\phi}_{29} = \left(\begin{array}{c}\displaystyle 12 z \left(- 9 x y^{2} z + 9 x y^{2} + 12 x y z - 12 x y - 3 x z + 3 x + 6 y^{2} z - 6 y^{2} - 8 y z + 8 y + 2 z - 2\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 1 of the reference element. $$\displaystyle l_{30}:v\mapsto\displaystyle\int_{f_{1}}(\left(\begin{array}{c}\displaystyle - s_{0}\\\displaystyle 0\\\displaystyle 0\end{array}\right))v$$
where $$f_{1}$$ is the 1st face.

$$\displaystyle \boldsymbol{\phi}_{30} = \left(\begin{array}{c}\displaystyle 12 z \left(9 x y^{2} z - 9 x y^{2} - 12 x y z + 12 x y + 3 x z - 3 x - 3 y^{2} z + 3 y^{2} + 4 y z - 4 y - z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 1 of the reference element. $$\displaystyle l_{31}:v\mapsto\displaystyle\int_{f_{1}}(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle s_{1}\end{array}\right))v$$
where $$f_{1}$$ is the 1st face.

$$\displaystyle \boldsymbol{\phi}_{31} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 12 x \left(- 9 x y^{2} z + 3 x y^{2} + 12 x y z - 4 x y - 3 x z + x + 9 y^{2} z - 3 y^{2} - 12 y z + 4 y + 3 z - 1\right)\end{array}\right)$$

This DOF is associated with face 1 of the reference element. $$\displaystyle l_{32}:v\mapsto\displaystyle\int_{f_{2}}(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1 - s_{1}\end{array}\right))v$$
where $$f_{2}$$ is the 2nd face.

$$\displaystyle \boldsymbol{\phi}_{32} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 12 y \left(9 x^{2} y z - 6 x^{2} y - 9 x^{2} z + 6 x^{2} - 12 x y z + 8 x y + 12 x z - 8 x + 3 y z - 2 y - 3 z + 2\right)\end{array}\right)$$

This DOF is associated with face 2 of the reference element. $$\displaystyle l_{33}:v\mapsto\displaystyle\int_{f_{2}}(\left(\begin{array}{c}\displaystyle 0\\\displaystyle s_{0} - 1\\\displaystyle 0\end{array}\right))v$$
where $$f_{2}$$ is the 2nd face.

$$\displaystyle \boldsymbol{\phi}_{33} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 12 z \left(- 9 x^{2} y z + 9 x^{2} y + 6 x^{2} z - 6 x^{2} + 12 x y z - 12 x y - 8 x z + 8 x - 3 y z + 3 y + 2 z - 2\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 2 of the reference element. $$\displaystyle l_{34}:v\mapsto\displaystyle\int_{f_{2}}(\left(\begin{array}{c}\displaystyle 0\\\displaystyle - s_{0}\\\displaystyle 0\end{array}\right))v$$
where $$f_{2}$$ is the 2nd face.

$$\displaystyle \boldsymbol{\phi}_{34} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 12 z \left(9 x^{2} y z - 9 x^{2} y - 3 x^{2} z + 3 x^{2} - 12 x y z + 12 x y + 4 x z - 4 x + 3 y z - 3 y - z + 1\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 2 of the reference element. $$\displaystyle l_{35}:v\mapsto\displaystyle\int_{f_{2}}(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle s_{1}\end{array}\right))v$$
where $$f_{2}$$ is the 2nd face.

$$\displaystyle \boldsymbol{\phi}_{35} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 12 y \left(- 9 x^{2} y z + 3 x^{2} y + 9 x^{2} z - 3 x^{2} + 12 x y z - 4 x y - 12 x z + 4 x - 3 y z + y + 3 z - 1\right)\end{array}\right)$$

This DOF is associated with face 2 of the reference element. $$\displaystyle l_{36}:v\mapsto\displaystyle\int_{f_{3}}(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1 - s_{1}\end{array}\right))v$$
where $$f_{3}$$ is the 3th face.

$$\displaystyle \boldsymbol{\phi}_{36} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 12 x y \left(9 x y z - 6 x y - 9 x z + 6 x - 6 y z + 4 y + 6 z - 4\right)\end{array}\right)$$

This DOF is associated with face 3 of the reference element. $$\displaystyle l_{37}:v\mapsto\displaystyle\int_{f_{3}}(\left(\begin{array}{c}\displaystyle 0\\\displaystyle s_{0} - 1\\\displaystyle 0\end{array}\right))v$$
where $$f_{3}$$ is the 3th face.

$$\displaystyle \boldsymbol{\phi}_{37} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 12 x z \left(- 9 x y z + 9 x y + 6 x z - 6 x + 6 y z - 6 y - 4 z + 4\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 3 of the reference element. $$\displaystyle l_{38}:v\mapsto\displaystyle\int_{f_{3}}(\left(\begin{array}{c}\displaystyle 0\\\displaystyle - s_{0}\\\displaystyle 0\end{array}\right))v$$
where $$f_{3}$$ is the 3th face.

$$\displaystyle \boldsymbol{\phi}_{38} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 12 x z \left(9 x y z - 9 x y - 3 x z + 3 x - 6 y z + 6 y + 2 z - 2\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 3 of the reference element. $$\displaystyle l_{39}:v\mapsto\displaystyle\int_{f_{3}}(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle s_{1}\end{array}\right))v$$
where $$f_{3}$$ is the 3th face.

$$\displaystyle \boldsymbol{\phi}_{39} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 12 x y \left(- 9 x y z + 3 x y + 9 x z - 3 x + 6 y z - 2 y - 6 z + 2\right)\end{array}\right)$$

This DOF is associated with face 3 of the reference element. $$\displaystyle l_{40}:v\mapsto\displaystyle\int_{f_{4}}(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1 - s_{1}\end{array}\right))v$$
where $$f_{4}$$ is the 4th face.

$$\displaystyle \boldsymbol{\phi}_{40} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 12 x y \left(9 x y z - 6 x y - 6 x z + 4 x - 9 y z + 6 y + 6 z - 4\right)\end{array}\right)$$

This DOF is associated with face 4 of the reference element. $$\displaystyle l_{41}:v\mapsto\displaystyle\int_{f_{4}}(\left(\begin{array}{c}\displaystyle s_{0} - 1\\\displaystyle 0\\\displaystyle 0\end{array}\right))v$$
where $$f_{4}$$ is the 4th face.

$$\displaystyle \boldsymbol{\phi}_{41} = \left(\begin{array}{c}\displaystyle 12 y z \left(- 9 x y z + 9 x y + 6 x z - 6 x + 6 y z - 6 y - 4 z + 4\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 4 of the reference element. $$\displaystyle l_{42}:v\mapsto\displaystyle\int_{f_{4}}(\left(\begin{array}{c}\displaystyle - s_{0}\\\displaystyle 0\\\displaystyle 0\end{array}\right))v$$
where $$f_{4}$$ is the 4th face.

$$\displaystyle \boldsymbol{\phi}_{42} = \left(\begin{array}{c}\displaystyle 12 y z \left(9 x y z - 9 x y - 6 x z + 6 x - 3 y z + 3 y + 2 z - 2\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 4 of the reference element. $$\displaystyle l_{43}:v\mapsto\displaystyle\int_{f_{4}}(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle s_{1}\end{array}\right))v$$
where $$f_{4}$$ is the 4th face.

$$\displaystyle \boldsymbol{\phi}_{43} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 12 x y \left(- 9 x y z + 3 x y + 6 x z - 2 x + 9 y z - 3 y - 6 z + 2\right)\end{array}\right)$$

This DOF is associated with face 4 of the reference element. $$\displaystyle l_{44}:v\mapsto\displaystyle\int_{f_{5}}(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1 - s_{1}\\\displaystyle 0\end{array}\right))v$$
where $$f_{5}$$ is the 5th face.

$$\displaystyle \boldsymbol{\phi}_{44} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 12 x z \left(9 x y z - 6 x y - 6 x z + 4 x - 9 y z + 6 y + 6 z - 4\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 5 of the reference element. $$\displaystyle l_{45}:v\mapsto\displaystyle\int_{f_{5}}(\left(\begin{array}{c}\displaystyle s_{0} - 1\\\displaystyle 0\\\displaystyle 0\end{array}\right))v$$
where $$f_{5}$$ is the 5th face.

$$\displaystyle \boldsymbol{\phi}_{45} = \left(\begin{array}{c}\displaystyle 12 y z \left(- 9 x y z + 6 x y + 9 x z - 6 x + 6 y z - 4 y - 6 z + 4\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 5 of the reference element. $$\displaystyle l_{46}:v\mapsto\displaystyle\int_{f_{5}}(\left(\begin{array}{c}\displaystyle - s_{0}\\\displaystyle 0\\\displaystyle 0\end{array}\right))v$$
where $$f_{5}$$ is the 5th face.

$$\displaystyle \boldsymbol{\phi}_{46} = \left(\begin{array}{c}\displaystyle 12 y z \left(9 x y z - 6 x y - 9 x z + 6 x - 3 y z + 2 y + 3 z - 2\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 5 of the reference element. $$\displaystyle l_{47}:v\mapsto\displaystyle\int_{f_{5}}(\left(\begin{array}{c}\displaystyle 0\\\displaystyle s_{1}\\\displaystyle 0\end{array}\right))v$$
where $$f_{5}$$ is the 5th face.

$$\displaystyle \boldsymbol{\phi}_{47} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 12 x z \left(- 9 x y z + 6 x y + 3 x z - 2 x + 9 y z - 6 y - 3 z + 2\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 5 of the reference element. $$\displaystyle l_{48}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1 - s_{2}\end{array}\right))v$$
where $$R$$ is the reference element.

$$\displaystyle \boldsymbol{\phi}_{48} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 72 x y \left(- 3 x y z + 2 x y + 3 x z - 2 x + 3 y z - 2 y - 3 z + 2\right)\end{array}\right)$$

This DOF is associated with volume 0 of the reference element. $$\displaystyle l_{49}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{c}\displaystyle 0\\\displaystyle s_{1} - 1\\\displaystyle 0\end{array}\right))v$$
where $$R$$ is the reference element.

$$\displaystyle \boldsymbol{\phi}_{49} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 72 x z \left(3 x y z - 3 x y - 2 x z + 2 x - 3 y z + 3 y + 2 z - 2\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with volume 0 of the reference element. $$\displaystyle l_{50}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{c}\displaystyle 1 - s_{0}\\\displaystyle 0\\\displaystyle 0\end{array}\right))v$$
where $$R$$ is the reference element.

$$\displaystyle \boldsymbol{\phi}_{50} = \left(\begin{array}{c}\displaystyle 72 y z \left(- 3 x y z + 3 x y + 3 x z - 3 x + 2 y z - 2 y - 2 z + 2\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

This DOF is associated with volume 0 of the reference element. $$\displaystyle l_{51}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{c}\displaystyle s_{0}\\\displaystyle 0\\\displaystyle 0\end{array}\right))v$$
where $$R$$ is the reference element.

$$\displaystyle \boldsymbol{\phi}_{51} = \left(\begin{array}{c}\displaystyle 72 y z \left(3 x y z - 3 x y - 3 x z + 3 x - y z + y + z - 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$

This DOF is associated with volume 0 of the reference element. $$\displaystyle l_{52}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{c}\displaystyle 0\\\displaystyle - s_{1}\\\displaystyle 0\end{array}\right))v$$
where $$R$$ is the reference element.

$$\displaystyle \boldsymbol{\phi}_{52} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 72 x z \left(- 3 x y z + 3 x y + x z - x + 3 y z - 3 y - z + 1\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with volume 0 of the reference element. $$\displaystyle l_{53}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle s_{2}\end{array}\right))v$$
where $$R$$ is the reference element.

$$\displaystyle \boldsymbol{\phi}_{53} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 72 x y \left(3 x y z - x y - 3 x z + x - 3 y z + y + 3 z - 1\right)\end{array}\right)$$

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