an encyclopedia of finite element definitions

# Wu–Xu

 Orders interval: $$k=2$$ triangle: $$k=3$$ tetrahedron: $$k=4$$ Reference elements interval, triangle, tetrahedron Polynomial set $$\mathcal{P}_{k-1} \oplus \mathcal{Z}^{(42)}_{k}$$ (interval) $$\mathcal{P}_{k-1} \oplus \mathcal{Z}^{(43)}_{k}$$ (triangle) $$\mathcal{P}_{k-1} \oplus \mathcal{Z}^{(44)}_{k}$$ (tetrahedron) ↓ Show polynomial set definitions ↓ DOFs On each vertex: point evaluations On each edge: integrals of normal derivatives On each face: integrals of normal derivatives On each volume: integrals of normal derivatives Number of DOFs interval: $$4$$triangle: $$12$$tetrahedron: $$38$$ Categories Scalar-valued elements

## Implementations

 Symfem "Wu-Xu"↓ Show Symfem examples ↓

## Examples

interval
order 2
triangle
order 3
tetrahedron
order 4
• $$R$$ is the reference interval. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$1$$, $$x$$, $$x^{2}$$, $$x^{2} \cdot \left(1 - x\right)$$
• $$\mathcal{L}=\{l_0,...,l_{3}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto v(0)$$

$$\displaystyle \phi_{0} = 2 x^{3} - 3 x^{2} + 1$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{1}:v\mapsto v'(0)$$

$$\displaystyle \phi_{1} = x \left(x^{2} - 2 x + 1\right)$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{2}:v\mapsto v(1)$$

$$\displaystyle \phi_{2} = x^{2} \cdot \left(3 - 2 x\right)$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{3}:v\mapsto v'(1)$$

$$\displaystyle \phi_{3} = x^{2} \left(x - 1\right)$$

This DOF is associated with vertex 1 of the reference element.
• $$R$$ is the reference triangle. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$1$$, $$x$$, $$x^{2}$$, $$x^{3}$$, $$y$$, $$x y$$, $$x^{2} y$$, $$y^{2}$$, $$x y^{2}$$, $$y^{3}$$, $$x^{2} y \left(- x - y + 1\right)$$, $$x y^{2} \left(- x - y + 1\right)$$
• $$\mathcal{L}=\{l_0,...,l_{11}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto v(0,0)$$

$$\displaystyle \phi_{0} = 12 x^{3} y + 2 x^{3} + 24 x^{2} y^{2} - 18 x^{2} y - 3 x^{2} + 12 x y^{3} - 18 x y^{2} + 6 x y + 2 y^{3} - 3 y^{2} + 1$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{1}:v\mapsto\frac{\partial}{\partial x}v(0,0)$$

$$\displaystyle \phi_{1} = x \left(- 4 x^{2} y + x^{2} + 6 x y - 2 x + 4 y^{3} - 3 y^{2} - 2 y + 1\right)$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{2}:v\mapsto\frac{\partial}{\partial y}v(0,0)$$

$$\displaystyle \phi_{2} = y \left(4 x^{3} - 3 x^{2} - 4 x y^{2} + 6 x y - 2 x + y^{2} - 2 y + 1\right)$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{3}:v\mapsto v(1,0)$$

$$\displaystyle \phi_{3} = x \left(- 6 x^{2} y - 2 x^{2} - 12 x y^{2} + 9 x y + 3 x - 6 y^{3} + 9 y^{2} - 3 y\right)$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{4}:v\mapsto\frac{\partial}{\partial x}v(1,0)$$

$$\displaystyle \phi_{4} = x^{2} \left(x - 1\right)$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{5}:v\mapsto\frac{\partial}{\partial y}v(1,0)$$

$$\displaystyle \phi_{5} = x y \left(- 4 x^{2} - 12 x y + 9 x - 8 y^{2} + 12 y - 4\right)$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{6}:v\mapsto v(0,1)$$

$$\displaystyle \phi_{6} = y \left(- 6 x^{3} - 12 x^{2} y + 9 x^{2} - 6 x y^{2} + 9 x y - 3 x - 2 y^{2} + 3 y\right)$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{7}:v\mapsto\frac{\partial}{\partial x}v(0,1)$$

$$\displaystyle \phi_{7} = x y \left(- 8 x^{2} - 12 x y + 12 x - 4 y^{2} + 9 y - 4\right)$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{8}:v\mapsto\frac{\partial}{\partial y}v(0,1)$$

$$\displaystyle \phi_{8} = y^{2} \left(y - 1\right)$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{9}:\mathbf{V}\mapsto\displaystyle\frac{\sqrt{2}}{2}\int_{e_{0}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle - \frac{\sqrt{2}}{2}\end{array}\right)}v$$
where $$e_{0}$$ is the 0th edge.

$$\displaystyle \phi_{9} = 3 \sqrt{2} x y \left(- 2 x^{2} - 4 x y + 3 x - 2 y^{2} + 3 y - 1\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{10}:\mathbf{V}\mapsto\displaystyle\int_{e_{1}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle -1\\\displaystyle 0\end{array}\right)}v$$
where $$e_{1}$$ is the 1st edge.

$$\displaystyle \phi_{10} = 6 x y \left(- 2 x^{2} - 2 x y + 3 x + y - 1\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{11}:\mathbf{V}\mapsto\displaystyle\int_{e_{2}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)}v$$
where $$e_{2}$$ is the 2nd edge.

$$\displaystyle \phi_{11} = 6 x y \left(2 x y - x + 2 y^{2} - 3 y + 1\right)$$

This DOF is associated with edge 2 of the reference element.
• $$R$$ is the reference tetrahedron. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$1$$, $$x$$, $$x^{2}$$, $$x^{3}$$, $$x^{4}$$, $$y$$, $$x y$$, $$x^{2} y$$, $$x^{3} y$$, $$y^{2}$$, $$x y^{2}$$, $$x^{2} y^{2}$$, $$y^{3}$$, $$x y^{3}$$, $$y^{4}$$, $$z$$, $$x z$$, $$x^{2} z$$, $$x^{3} z$$, $$y z$$, $$x y z$$, $$x^{2} y z$$, $$y^{2} z$$, $$x y^{2} z$$, $$y^{3} z$$, $$z^{2}$$, $$x z^{2}$$, $$x^{2} z^{2}$$, $$y z^{2}$$, $$x y z^{2}$$, $$y^{2} z^{2}$$, $$z^{3}$$, $$x z^{3}$$, $$y z^{3}$$, $$z^{4}$$, $$x^{2} y z \left(- x - y - z + 1\right)$$, $$x y^{2} z \left(- x - y - z + 1\right)$$, $$x y z^{2} \left(- x - y - z + 1\right)$$
• $$\mathcal{L}=\{l_0,...,l_{37}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto v(0,0,0)$$

$$\displaystyle \phi_{0} = - \frac{3 x^{4}}{7} - \frac{144 x^{3} y z}{7} - \frac{48 x^{3} y}{7} - \frac{48 x^{3} z}{7} + \frac{20 x^{3}}{7} - \frac{288 x^{2} y^{2} z}{7} - \frac{54 x^{2} y^{2}}{7} - \frac{288 x^{2} y z^{2}}{7} + \frac{144 x^{2} y z}{7} + 12 x^{2} y - \frac{54 x^{2} z^{2}}{7} + 12 x^{2} z - \frac{24 x^{2}}{7} - \frac{144 x y^{3} z}{7} - \frac{48 x y^{3}}{7} - \frac{288 x y^{2} z^{2}}{7} + \frac{144 x y^{2} z}{7} + 12 x y^{2} - \frac{144 x y z^{3}}{7} + \frac{144 x y z^{2}}{7} + \frac{48 x y z}{7} - \frac{36 x y}{7} - \frac{48 x z^{3}}{7} + 12 x z^{2} - \frac{36 x z}{7} - \frac{3 y^{4}}{7} - \frac{48 y^{3} z}{7} + \frac{20 y^{3}}{7} - \frac{54 y^{2} z^{2}}{7} + 12 y^{2} z - \frac{24 y^{2}}{7} - \frac{48 y z^{3}}{7} + 12 y z^{2} - \frac{36 y z}{7} - \frac{3 z^{4}}{7} + \frac{20 z^{3}}{7} - \frac{24 z^{2}}{7} + 1$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{1}:v\mapsto\frac{\partial}{\partial x}v(0,0,0)$$

$$\displaystyle \phi_{1} = - \frac{229 x^{4}}{308} + \frac{1074 x^{3} y z}{77} - \frac{691 x^{3} y}{154} - \frac{691 x^{3} z}{154} + \frac{383 x^{3}}{154} + \frac{636 x^{2} y^{2} z}{77} - \frac{549 x^{2} y^{2}}{154} + \frac{636 x^{2} y z^{2}}{77} - \frac{3807 x^{2} y z}{154} + \frac{173 x^{2} y}{22} - \frac{549 x^{2} z^{2}}{154} + \frac{173 x^{2} z}{22} - \frac{845 x^{2}}{308} - \frac{438 x y^{3} z}{77} + \frac{65 x y^{3}}{154} - \frac{876 x y^{2} z^{2}}{77} + \frac{435 x y^{2} z}{154} + \frac{43 x y^{2}}{22} - \frac{438 x y z^{3}}{77} + \frac{435 x y z^{2}}{154} + \frac{54 x y z}{7} - \frac{260 x y}{77} + \frac{65 x z^{3}}{154} + \frac{43 x z^{2}}{22} - \frac{260 x z}{77} + x + \frac{65 y^{4}}{308} + \frac{65 y^{3} z}{154} - \frac{65 y^{3}}{154} + \frac{207 y^{2} z^{2}}{154} - \frac{29 y^{2} z}{22} + \frac{65 y^{2}}{308} + \frac{65 y z^{3}}{154} - \frac{29 y z^{2}}{22} + \frac{69 y z}{77} + \frac{65 z^{4}}{308} - \frac{65 z^{3}}{154} + \frac{65 z^{2}}{308}$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{2}:v\mapsto\frac{\partial}{\partial y}v(0,0,0)$$

$$\displaystyle \phi_{2} = \frac{65 x^{4}}{308} - \frac{438 x^{3} y z}{77} + \frac{65 x^{3} y}{154} + \frac{65 x^{3} z}{154} - \frac{65 x^{3}}{154} + \frac{636 x^{2} y^{2} z}{77} - \frac{549 x^{2} y^{2}}{154} - \frac{876 x^{2} y z^{2}}{77} + \frac{435 x^{2} y z}{154} + \frac{43 x^{2} y}{22} + \frac{207 x^{2} z^{2}}{154} - \frac{29 x^{2} z}{22} + \frac{65 x^{2}}{308} + \frac{1074 x y^{3} z}{77} - \frac{691 x y^{3}}{154} + \frac{636 x y^{2} z^{2}}{77} - \frac{3807 x y^{2} z}{154} + \frac{173 x y^{2}}{22} - \frac{438 x y z^{3}}{77} + \frac{435 x y z^{2}}{154} + \frac{54 x y z}{7} - \frac{260 x y}{77} + \frac{65 x z^{3}}{154} - \frac{29 x z^{2}}{22} + \frac{69 x z}{77} - \frac{229 y^{4}}{308} - \frac{691 y^{3} z}{154} + \frac{383 y^{3}}{154} - \frac{549 y^{2} z^{2}}{154} + \frac{173 y^{2} z}{22} - \frac{845 y^{2}}{308} + \frac{65 y z^{3}}{154} + \frac{43 y z^{2}}{22} - \frac{260 y z}{77} + y + \frac{65 z^{4}}{308} - \frac{65 z^{3}}{154} + \frac{65 z^{2}}{308}$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{3}:v\mapsto\frac{\partial}{\partial z}v(0,0,0)$$

$$\displaystyle \phi_{3} = \frac{65 x^{4}}{308} - \frac{438 x^{3} y z}{77} + \frac{65 x^{3} y}{154} + \frac{65 x^{3} z}{154} - \frac{65 x^{3}}{154} - \frac{876 x^{2} y^{2} z}{77} + \frac{207 x^{2} y^{2}}{154} + \frac{636 x^{2} y z^{2}}{77} + \frac{435 x^{2} y z}{154} - \frac{29 x^{2} y}{22} - \frac{549 x^{2} z^{2}}{154} + \frac{43 x^{2} z}{22} + \frac{65 x^{2}}{308} - \frac{438 x y^{3} z}{77} + \frac{65 x y^{3}}{154} + \frac{636 x y^{2} z^{2}}{77} + \frac{435 x y^{2} z}{154} - \frac{29 x y^{2}}{22} + \frac{1074 x y z^{3}}{77} - \frac{3807 x y z^{2}}{154} + \frac{54 x y z}{7} + \frac{69 x y}{77} - \frac{691 x z^{3}}{154} + \frac{173 x z^{2}}{22} - \frac{260 x z}{77} + \frac{65 y^{4}}{308} + \frac{65 y^{3} z}{154} - \frac{65 y^{3}}{154} - \frac{549 y^{2} z^{2}}{154} + \frac{43 y^{2} z}{22} + \frac{65 y^{2}}{308} - \frac{691 y z^{3}}{154} + \frac{173 y z^{2}}{22} - \frac{260 y z}{77} - \frac{229 z^{4}}{308} + \frac{383 z^{3}}{154} - \frac{845 z^{2}}{308} + z$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{4}:v\mapsto v(1,0,0)$$

$$\displaystyle \phi_{4} = - \frac{13 x^{4}}{7} + \frac{48 x^{3} y z}{7} + \frac{16 x^{3} y}{7} + \frac{16 x^{3} z}{7} + \frac{12 x^{3}}{7} + \frac{96 x^{2} y^{2} z}{7} + \frac{18 x^{2} y^{2}}{7} + \frac{96 x^{2} y z^{2}}{7} - \frac{48 x^{2} y z}{7} - 4 x^{2} y + \frac{18 x^{2} z^{2}}{7} - 4 x^{2} z + \frac{8 x^{2}}{7} + \frac{48 x y^{3} z}{7} + \frac{16 x y^{3}}{7} + \frac{96 x y^{2} z^{2}}{7} - \frac{48 x y^{2} z}{7} - 4 x y^{2} + \frac{48 x y z^{3}}{7} - \frac{48 x y z^{2}}{7} - \frac{16 x y z}{7} + \frac{12 x y}{7} + \frac{16 x z^{3}}{7} - 4 x z^{2} + \frac{12 x z}{7} + \frac{8 y^{4}}{7} + \frac{16 y^{3} z}{7} - \frac{16 y^{3}}{7} + \frac{18 y^{2} z^{2}}{7} - 4 y^{2} z + \frac{8 y^{2}}{7} + \frac{16 y z^{3}}{7} - 4 y z^{2} + \frac{12 y z}{7} + \frac{8 z^{4}}{7} - \frac{16 z^{3}}{7} + \frac{8 z^{2}}{7}$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{5}:v\mapsto\frac{\partial}{\partial x}v(1,0,0)$$

$$\displaystyle \phi_{5} = \frac{26 x^{4}}{33} + \frac{4 x^{3} y z}{11} - \frac{89 x^{3} y}{132} - \frac{89 x^{3} z}{132} - \frac{19 x^{3}}{33} + \frac{2 x^{2} y^{2} z}{11} - \frac{81 x^{2} y^{2}}{88} + \frac{2 x^{2} y z^{2}}{11} - \frac{16 x^{2} y z}{11} + \frac{137 x^{2} y}{132} - \frac{81 x^{2} z^{2}}{88} + \frac{137 x^{2} z}{132} - \frac{7 x^{2}}{33} - \frac{2 x y^{3} z}{11} - \frac{71 x y^{3}}{132} - \frac{4 x y^{2} z^{2}}{11} - \frac{17 x y^{2} z}{22} + \frac{119 x y^{2}}{132} - \frac{2 x y z^{3}}{11} - \frac{17 x y z^{2}}{22} + \frac{4 x y z}{3} - \frac{4 x y}{11} - \frac{71 x z^{3}}{132} + \frac{119 x z^{2}}{132} - \frac{4 x z}{11} - \frac{19 y^{4}}{132} - \frac{19 y^{3} z}{66} + \frac{19 y^{3}}{66} - \frac{9 y^{2} z^{2}}{22} + \frac{37 y^{2} z}{66} - \frac{19 y^{2}}{132} - \frac{19 y z^{3}}{66} + \frac{37 y z^{2}}{66} - \frac{3 y z}{11} - \frac{19 z^{4}}{132} + \frac{19 z^{3}}{66} - \frac{19 z^{2}}{132}$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{6}:v\mapsto\frac{\partial}{\partial y}v(1,0,0)$$

$$\displaystyle \phi_{6} = \frac{283 x^{4}}{1848} + \frac{265 x^{3} y z}{77} + \frac{257 x^{3} y}{462} + \frac{283 x^{3} z}{924} - \frac{283 x^{3}}{924} + \frac{950 x^{2} y^{2} z}{77} - \frac{1053 x^{2} y^{2}}{616} + \frac{404 x^{2} y z^{2}}{77} - \frac{2425 x^{2} y z}{308} + \frac{5 x^{2} y}{6} + \frac{135 x^{2} z^{2}}{308} - \frac{79 x^{2} z}{132} + \frac{283 x^{2}}{1848} + \frac{685 x y^{3} z}{77} - \frac{302 x y^{3}}{231} + \frac{824 x y^{2} z^{2}}{77} - \frac{4063 x y^{2} z}{308} + \frac{56 x y^{2}}{33} + \frac{139 x y z^{3}}{77} - \frac{1795 x y z^{2}}{308} + \frac{79 x y z}{21} - \frac{30 x y}{77} + \frac{661 x z^{3}}{924} - \frac{133 x z^{2}}{132} + \frac{45 x z}{154} - \frac{515 y^{4}}{1848} - \frac{515 y^{3} z}{924} + \frac{515 y^{3}}{924} - \frac{27 y^{2} z^{2}}{154} + \frac{89 y^{2} z}{132} - \frac{515 y^{2}}{1848} + \frac{661 y z^{3}}{924} - \frac{79 y z^{2}}{132} - \frac{9 y z}{77} + \frac{661 z^{4}}{1848} - \frac{661 z^{3}}{924} + \frac{661 z^{2}}{1848}$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{7}:v\mapsto\frac{\partial}{\partial z}v(1,0,0)$$

$$\displaystyle \phi_{7} = \frac{283 x^{4}}{1848} + \frac{265 x^{3} y z}{77} + \frac{283 x^{3} y}{924} + \frac{257 x^{3} z}{462} - \frac{283 x^{3}}{924} + \frac{404 x^{2} y^{2} z}{77} + \frac{135 x^{2} y^{2}}{308} + \frac{950 x^{2} y z^{2}}{77} - \frac{2425 x^{2} y z}{308} - \frac{79 x^{2} y}{132} - \frac{1053 x^{2} z^{2}}{616} + \frac{5 x^{2} z}{6} + \frac{283 x^{2}}{1848} + \frac{139 x y^{3} z}{77} + \frac{661 x y^{3}}{924} + \frac{824 x y^{2} z^{2}}{77} - \frac{1795 x y^{2} z}{308} - \frac{133 x y^{2}}{132} + \frac{685 x y z^{3}}{77} - \frac{4063 x y z^{2}}{308} + \frac{79 x y z}{21} + \frac{45 x y}{154} - \frac{302 x z^{3}}{231} + \frac{56 x z^{2}}{33} - \frac{30 x z}{77} + \frac{661 y^{4}}{1848} + \frac{661 y^{3} z}{924} - \frac{661 y^{3}}{924} - \frac{27 y^{2} z^{2}}{154} - \frac{79 y^{2} z}{132} + \frac{661 y^{2}}{1848} - \frac{515 y z^{3}}{924} + \frac{89 y z^{2}}{132} - \frac{9 y z}{77} - \frac{515 z^{4}}{1848} + \frac{515 z^{3}}{924} - \frac{515 z^{2}}{1848}$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{8}:v\mapsto v(0,1,0)$$

$$\displaystyle \phi_{8} = \frac{8 x^{4}}{7} + \frac{48 x^{3} y z}{7} + \frac{16 x^{3} y}{7} + \frac{16 x^{3} z}{7} - \frac{16 x^{3}}{7} + \frac{96 x^{2} y^{2} z}{7} + \frac{18 x^{2} y^{2}}{7} + \frac{96 x^{2} y z^{2}}{7} - \frac{48 x^{2} y z}{7} - 4 x^{2} y + \frac{18 x^{2} z^{2}}{7} - 4 x^{2} z + \frac{8 x^{2}}{7} + \frac{48 x y^{3} z}{7} + \frac{16 x y^{3}}{7} + \frac{96 x y^{2} z^{2}}{7} - \frac{48 x y^{2} z}{7} - 4 x y^{2} + \frac{48 x y z^{3}}{7} - \frac{48 x y z^{2}}{7} - \frac{16 x y z}{7} + \frac{12 x y}{7} + \frac{16 x z^{3}}{7} - 4 x z^{2} + \frac{12 x z}{7} - \frac{13 y^{4}}{7} + \frac{16 y^{3} z}{7} + \frac{12 y^{3}}{7} + \frac{18 y^{2} z^{2}}{7} - 4 y^{2} z + \frac{8 y^{2}}{7} + \frac{16 y z^{3}}{7} - 4 y z^{2} + \frac{12 y z}{7} + \frac{8 z^{4}}{7} - \frac{16 z^{3}}{7} + \frac{8 z^{2}}{7}$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{9}:v\mapsto\frac{\partial}{\partial x}v(0,1,0)$$

$$\displaystyle \phi_{9} = - \frac{515 x^{4}}{1848} + \frac{685 x^{3} y z}{77} - \frac{302 x^{3} y}{231} - \frac{515 x^{3} z}{924} + \frac{515 x^{3}}{924} + \frac{950 x^{2} y^{2} z}{77} - \frac{1053 x^{2} y^{2}}{616} + \frac{824 x^{2} y z^{2}}{77} - \frac{4063 x^{2} y z}{308} + \frac{56 x^{2} y}{33} - \frac{27 x^{2} z^{2}}{154} + \frac{89 x^{2} z}{132} - \frac{515 x^{2}}{1848} + \frac{265 x y^{3} z}{77} + \frac{257 x y^{3}}{462} + \frac{404 x y^{2} z^{2}}{77} - \frac{2425 x y^{2} z}{308} + \frac{5 x y^{2}}{6} + \frac{139 x y z^{3}}{77} - \frac{1795 x y z^{2}}{308} + \frac{79 x y z}{21} - \frac{30 x y}{77} + \frac{661 x z^{3}}{924} - \frac{79 x z^{2}}{132} - \frac{9 x z}{77} + \frac{283 y^{4}}{1848} + \frac{283 y^{3} z}{924} - \frac{283 y^{3}}{924} + \frac{135 y^{2} z^{2}}{308} - \frac{79 y^{2} z}{132} + \frac{283 y^{2}}{1848} + \frac{661 y z^{3}}{924} - \frac{133 y z^{2}}{132} + \frac{45 y z}{154} + \frac{661 z^{4}}{1848} - \frac{661 z^{3}}{924} + \frac{661 z^{2}}{1848}$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{10}:v\mapsto\frac{\partial}{\partial y}v(0,1,0)$$

$$\displaystyle \phi_{10} = - \frac{19 x^{4}}{132} - \frac{2 x^{3} y z}{11} - \frac{71 x^{3} y}{132} - \frac{19 x^{3} z}{66} + \frac{19 x^{3}}{66} + \frac{2 x^{2} y^{2} z}{11} - \frac{81 x^{2} y^{2}}{88} - \frac{4 x^{2} y z^{2}}{11} - \frac{17 x^{2} y z}{22} + \frac{119 x^{2} y}{132} - \frac{9 x^{2} z^{2}}{22} + \frac{37 x^{2} z}{66} - \frac{19 x^{2}}{132} + \frac{4 x y^{3} z}{11} - \frac{89 x y^{3}}{132} + \frac{2 x y^{2} z^{2}}{11} - \frac{16 x y^{2} z}{11} + \frac{137 x y^{2}}{132} - \frac{2 x y z^{3}}{11} - \frac{17 x y z^{2}}{22} + \frac{4 x y z}{3} - \frac{4 x y}{11} - \frac{19 x z^{3}}{66} + \frac{37 x z^{2}}{66} - \frac{3 x z}{11} + \frac{26 y^{4}}{33} - \frac{89 y^{3} z}{132} - \frac{19 y^{3}}{33} - \frac{81 y^{2} z^{2}}{88} + \frac{137 y^{2} z}{132} - \frac{7 y^{2}}{33} - \frac{71 y z^{3}}{132} + \frac{119 y z^{2}}{132} - \frac{4 y z}{11} - \frac{19 z^{4}}{132} + \frac{19 z^{3}}{66} - \frac{19 z^{2}}{132}$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{11}:v\mapsto\frac{\partial}{\partial z}v(0,1,0)$$

$$\displaystyle \phi_{11} = \frac{661 x^{4}}{1848} + \frac{139 x^{3} y z}{77} + \frac{661 x^{3} y}{924} + \frac{661 x^{3} z}{924} - \frac{661 x^{3}}{924} + \frac{404 x^{2} y^{2} z}{77} + \frac{135 x^{2} y^{2}}{308} + \frac{824 x^{2} y z^{2}}{77} - \frac{1795 x^{2} y z}{308} - \frac{133 x^{2} y}{132} - \frac{27 x^{2} z^{2}}{154} - \frac{79 x^{2} z}{132} + \frac{661 x^{2}}{1848} + \frac{265 x y^{3} z}{77} + \frac{283 x y^{3}}{924} + \frac{950 x y^{2} z^{2}}{77} - \frac{2425 x y^{2} z}{308} - \frac{79 x y^{2}}{132} + \frac{685 x y z^{3}}{77} - \frac{4063 x y z^{2}}{308} + \frac{79 x y z}{21} + \frac{45 x y}{154} - \frac{515 x z^{3}}{924} + \frac{89 x z^{2}}{132} - \frac{9 x z}{77} + \frac{283 y^{4}}{1848} + \frac{257 y^{3} z}{462} - \frac{283 y^{3}}{924} - \frac{1053 y^{2} z^{2}}{616} + \frac{5 y^{2} z}{6} + \frac{283 y^{2}}{1848} - \frac{302 y z^{3}}{231} + \frac{56 y z^{2}}{33} - \frac{30 y z}{77} - \frac{515 z^{4}}{1848} + \frac{515 z^{3}}{924} - \frac{515 z^{2}}{1848}$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{12}:v\mapsto v(0,0,1)$$

$$\displaystyle \phi_{12} = \frac{8 x^{4}}{7} + \frac{48 x^{3} y z}{7} + \frac{16 x^{3} y}{7} + \frac{16 x^{3} z}{7} - \frac{16 x^{3}}{7} + \frac{96 x^{2} y^{2} z}{7} + \frac{18 x^{2} y^{2}}{7} + \frac{96 x^{2} y z^{2}}{7} - \frac{48 x^{2} y z}{7} - 4 x^{2} y + \frac{18 x^{2} z^{2}}{7} - 4 x^{2} z + \frac{8 x^{2}}{7} + \frac{48 x y^{3} z}{7} + \frac{16 x y^{3}}{7} + \frac{96 x y^{2} z^{2}}{7} - \frac{48 x y^{2} z}{7} - 4 x y^{2} + \frac{48 x y z^{3}}{7} - \frac{48 x y z^{2}}{7} - \frac{16 x y z}{7} + \frac{12 x y}{7} + \frac{16 x z^{3}}{7} - 4 x z^{2} + \frac{12 x z}{7} + \frac{8 y^{4}}{7} + \frac{16 y^{3} z}{7} - \frac{16 y^{3}}{7} + \frac{18 y^{2} z^{2}}{7} - 4 y^{2} z + \frac{8 y^{2}}{7} + \frac{16 y z^{3}}{7} - 4 y z^{2} + \frac{12 y z}{7} - \frac{13 z^{4}}{7} + \frac{12 z^{3}}{7} + \frac{8 z^{2}}{7}$$

This DOF is associated with vertex 3 of the reference element.
$$\displaystyle l_{13}:v\mapsto\frac{\partial}{\partial x}v(0,0,1)$$

$$\displaystyle \phi_{13} = - \frac{515 x^{4}}{1848} + \frac{685 x^{3} y z}{77} - \frac{515 x^{3} y}{924} - \frac{302 x^{3} z}{231} + \frac{515 x^{3}}{924} + \frac{824 x^{2} y^{2} z}{77} - \frac{27 x^{2} y^{2}}{154} + \frac{950 x^{2} y z^{2}}{77} - \frac{4063 x^{2} y z}{308} + \frac{89 x^{2} y}{132} - \frac{1053 x^{2} z^{2}}{616} + \frac{56 x^{2} z}{33} - \frac{515 x^{2}}{1848} + \frac{139 x y^{3} z}{77} + \frac{661 x y^{3}}{924} + \frac{404 x y^{2} z^{2}}{77} - \frac{1795 x y^{2} z}{308} - \frac{79 x y^{2}}{132} + \frac{265 x y z^{3}}{77} - \frac{2425 x y z^{2}}{308} + \frac{79 x y z}{21} - \frac{9 x y}{77} + \frac{257 x z^{3}}{462} + \frac{5 x z^{2}}{6} - \frac{30 x z}{77} + \frac{661 y^{4}}{1848} + \frac{661 y^{3} z}{924} - \frac{661 y^{3}}{924} + \frac{135 y^{2} z^{2}}{308} - \frac{133 y^{2} z}{132} + \frac{661 y^{2}}{1848} + \frac{283 y z^{3}}{924} - \frac{79 y z^{2}}{132} + \frac{45 y z}{154} + \frac{283 z^{4}}{1848} - \frac{283 z^{3}}{924} + \frac{283 z^{2}}{1848}$$

This DOF is associated with vertex 3 of the reference element.
$$\displaystyle l_{14}:v\mapsto\frac{\partial}{\partial y}v(0,0,1)$$

$$\displaystyle \phi_{14} = \frac{661 x^{4}}{1848} + \frac{139 x^{3} y z}{77} + \frac{661 x^{3} y}{924} + \frac{661 x^{3} z}{924} - \frac{661 x^{3}}{924} + \frac{824 x^{2} y^{2} z}{77} - \frac{27 x^{2} y^{2}}{154} + \frac{404 x^{2} y z^{2}}{77} - \frac{1795 x^{2} y z}{308} - \frac{79 x^{2} y}{132} + \frac{135 x^{2} z^{2}}{308} - \frac{133 x^{2} z}{132} + \frac{661 x^{2}}{1848} + \frac{685 x y^{3} z}{77} - \frac{515 x y^{3}}{924} + \frac{950 x y^{2} z^{2}}{77} - \frac{4063 x y^{2} z}{308} + \frac{89 x y^{2}}{132} + \frac{265 x y z^{3}}{77} - \frac{2425 x y z^{2}}{308} + \frac{79 x y z}{21} - \frac{9 x y}{77} + \frac{283 x z^{3}}{924} - \frac{79 x z^{2}}{132} + \frac{45 x z}{154} - \frac{515 y^{4}}{1848} - \frac{302 y^{3} z}{231} + \frac{515 y^{3}}{924} - \frac{1053 y^{2} z^{2}}{616} + \frac{56 y^{2} z}{33} - \frac{515 y^{2}}{1848} + \frac{257 y z^{3}}{462} + \frac{5 y z^{2}}{6} - \frac{30 y z}{77} + \frac{283 z^{4}}{1848} - \frac{283 z^{3}}{924} + \frac{283 z^{2}}{1848}$$

This DOF is associated with vertex 3 of the reference element.
$$\displaystyle l_{15}:v\mapsto\frac{\partial}{\partial z}v(0,0,1)$$

$$\displaystyle \phi_{15} = - \frac{19 x^{4}}{132} - \frac{2 x^{3} y z}{11} - \frac{19 x^{3} y}{66} - \frac{71 x^{3} z}{132} + \frac{19 x^{3}}{66} - \frac{4 x^{2} y^{2} z}{11} - \frac{9 x^{2} y^{2}}{22} + \frac{2 x^{2} y z^{2}}{11} - \frac{17 x^{2} y z}{22} + \frac{37 x^{2} y}{66} - \frac{81 x^{2} z^{2}}{88} + \frac{119 x^{2} z}{132} - \frac{19 x^{2}}{132} - \frac{2 x y^{3} z}{11} - \frac{19 x y^{3}}{66} + \frac{2 x y^{2} z^{2}}{11} - \frac{17 x y^{2} z}{22} + \frac{37 x y^{2}}{66} + \frac{4 x y z^{3}}{11} - \frac{16 x y z^{2}}{11} + \frac{4 x y z}{3} - \frac{3 x y}{11} - \frac{89 x z^{3}}{132} + \frac{137 x z^{2}}{132} - \frac{4 x z}{11} - \frac{19 y^{4}}{132} - \frac{71 y^{3} z}{132} + \frac{19 y^{3}}{66} - \frac{81 y^{2} z^{2}}{88} + \frac{119 y^{2} z}{132} - \frac{19 y^{2}}{132} - \frac{89 y z^{3}}{132} + \frac{137 y z^{2}}{132} - \frac{4 y z}{11} + \frac{26 z^{4}}{33} - \frac{19 z^{3}}{33} - \frac{7 z^{2}}{33}$$

This DOF is associated with vertex 3 of the reference element.
$$\displaystyle l_{16}:\mathbf{V}\mapsto\displaystyle\frac{\sqrt{3}}{3}\int_{f_{0}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle \frac{\sqrt{3}}{3}\\\displaystyle \frac{\sqrt{3}}{3}\\\displaystyle \frac{\sqrt{3}}{3}\end{array}\right)}v$$
where $$f_{0}$$ is the 0th face.

$$\displaystyle \phi_{16} = \frac{5 \sqrt{3} \left(- 4 x^{4} - 24 x^{3} y z - 8 x^{3} y - 8 x^{3} z + 8 x^{3} - 48 x^{2} y^{2} z - 9 x^{2} y^{2} - 48 x^{2} y z^{2} + 24 x^{2} y z + 14 x^{2} y - 9 x^{2} z^{2} + 14 x^{2} z - 4 x^{2} - 24 x y^{3} z - 8 x y^{3} - 48 x y^{2} z^{2} + 24 x y^{2} z + 14 x y^{2} - 24 x y z^{3} + 24 x y z^{2} + 8 x y z - 6 x y - 8 x z^{3} + 14 x z^{2} - 6 x z - 4 y^{4} - 8 y^{3} z + 8 y^{3} - 9 y^{2} z^{2} + 14 y^{2} z - 4 y^{2} - 8 y z^{3} + 14 y z^{2} - 6 y z - 4 z^{4} + 8 z^{3} - 4 z^{2}\right)}{7}$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{17}:\mathbf{V}\mapsto\displaystyle\int_{f_{1}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)}v$$
where $$f_{1}$$ is the 1st face.

$$\displaystyle \phi_{17} = \frac{585 x^{4}}{77} - \frac{4680 x^{3} y z}{77} + \frac{1170 x^{3} y}{77} + \frac{1170 x^{3} z}{77} - \frac{1170 x^{3}}{77} - \frac{4320 x^{2} y^{2} z}{77} + \frac{1080 x^{2} y^{2}}{77} - \frac{4320 x^{2} y z^{2}}{77} + \frac{7830 x^{2} y z}{77} - \frac{270 x^{2} y}{11} + \frac{1080 x^{2} z^{2}}{77} - \frac{270 x^{2} z}{11} + \frac{585 x^{2}}{77} + \frac{360 x y^{3} z}{77} - \frac{90 x y^{3}}{77} + \frac{720 x y^{2} z^{2}}{77} + \frac{1530 x y^{2} z}{77} - \frac{90 x y^{2}}{11} + \frac{360 x y z^{3}}{77} + \frac{1530 x y z^{2}}{77} - \frac{240 x y z}{7} + \frac{720 x y}{77} - \frac{90 x z^{3}}{77} - \frac{90 x z^{2}}{11} + \frac{720 x z}{77} - \frac{45 y^{4}}{77} - \frac{90 y^{3} z}{77} + \frac{90 y^{3}}{77} - \frac{180 y^{2} z^{2}}{77} + \frac{30 y^{2} z}{11} - \frac{45 y^{2}}{77} - \frac{90 y z^{3}}{77} + \frac{30 y z^{2}}{11} - \frac{120 y z}{77} - \frac{45 z^{4}}{77} + \frac{90 z^{3}}{77} - \frac{45 z^{2}}{77}$$

This DOF is associated with face 1 of the reference element.
$$\displaystyle l_{18}:\mathbf{V}\mapsto\displaystyle\int_{f_{2}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle 0\\\displaystyle -1\\\displaystyle 0\end{array}\right)}v$$
where $$f_{2}$$ is the 2nd face.

$$\displaystyle \phi_{18} = \frac{45 x^{4}}{77} - \frac{360 x^{3} y z}{77} + \frac{90 x^{3} y}{77} + \frac{90 x^{3} z}{77} - \frac{90 x^{3}}{77} + \frac{4320 x^{2} y^{2} z}{77} - \frac{1080 x^{2} y^{2}}{77} - \frac{720 x^{2} y z^{2}}{77} - \frac{1530 x^{2} y z}{77} + \frac{90 x^{2} y}{11} + \frac{180 x^{2} z^{2}}{77} - \frac{30 x^{2} z}{11} + \frac{45 x^{2}}{77} + \frac{4680 x y^{3} z}{77} - \frac{1170 x y^{3}}{77} + \frac{4320 x y^{2} z^{2}}{77} - \frac{7830 x y^{2} z}{77} + \frac{270 x y^{2}}{11} - \frac{360 x y z^{3}}{77} - \frac{1530 x y z^{2}}{77} + \frac{240 x y z}{7} - \frac{720 x y}{77} + \frac{90 x z^{3}}{77} - \frac{30 x z^{2}}{11} + \frac{120 x z}{77} - \frac{585 y^{4}}{77} - \frac{1170 y^{3} z}{77} + \frac{1170 y^{3}}{77} - \frac{1080 y^{2} z^{2}}{77} + \frac{270 y^{2} z}{11} - \frac{585 y^{2}}{77} + \frac{90 y z^{3}}{77} + \frac{90 y z^{2}}{11} - \frac{720 y z}{77} + \frac{45 z^{4}}{77} - \frac{90 z^{3}}{77} + \frac{45 z^{2}}{77}$$

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{19}:\mathbf{V}\mapsto\displaystyle\int_{f_{3}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)}v$$
where $$f_{3}$$ is the 3th face.

$$\displaystyle \phi_{19} = - \frac{45 x^{4}}{77} + \frac{360 x^{3} y z}{77} - \frac{90 x^{3} y}{77} - \frac{90 x^{3} z}{77} + \frac{90 x^{3}}{77} + \frac{720 x^{2} y^{2} z}{77} - \frac{180 x^{2} y^{2}}{77} - \frac{4320 x^{2} y z^{2}}{77} + \frac{1530 x^{2} y z}{77} + \frac{30 x^{2} y}{11} + \frac{1080 x^{2} z^{2}}{77} - \frac{90 x^{2} z}{11} - \frac{45 x^{2}}{77} + \frac{360 x y^{3} z}{77} - \frac{90 x y^{3}}{77} - \frac{4320 x y^{2} z^{2}}{77} + \frac{1530 x y^{2} z}{77} + \frac{30 x y^{2}}{11} - \frac{4680 x y z^{3}}{77} + \frac{7830 x y z^{2}}{77} - \frac{240 x y z}{7} - \frac{120 x y}{77} + \frac{1170 x z^{3}}{77} - \frac{270 x z^{2}}{11} + \frac{720 x z}{77} - \frac{45 y^{4}}{77} - \frac{90 y^{3} z}{77} + \frac{90 y^{3}}{77} + \frac{1080 y^{2} z^{2}}{77} - \frac{90 y^{2} z}{11} - \frac{45 y^{2}}{77} + \frac{1170 y z^{3}}{77} - \frac{270 y z^{2}}{11} + \frac{720 y z}{77} + \frac{585 z^{4}}{77} - \frac{1170 z^{3}}{77} + \frac{585 z^{2}}{77}$$

This DOF is associated with face 3 of the reference element.
$$\displaystyle l_{20}:\mathbf{V}\mapsto\displaystyle\frac{\sqrt{2}}{2}\int_{e_{0}}\frac{\partial^{2}}{\partial\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)^{2}}v$$
where $$e_{0}$$ is the 0th edge.

$$\displaystyle \phi_{20} = - \frac{149 x^{4}}{616} + \frac{303 x^{3} y z}{77} - \frac{149 x^{3} y}{308} - \frac{149 x^{3} z}{308} + \frac{149 x^{3}}{308} + \frac{564 x^{2} y^{2} z}{77} + \frac{129 x^{2} y^{2}}{308} + \frac{564 x^{2} y z^{2}}{77} - \frac{1779 x^{2} y z}{308} + \frac{9 x^{2} y}{44} + \frac{129 x^{2} z^{2}}{308} + \frac{9 x^{2} z}{44} - \frac{149 x^{2}}{616} + \frac{261 x y^{3} z}{77} + \frac{47 x y^{3}}{308} + \frac{522 x y^{2} z^{2}}{77} - \frac{2031 x y^{2} z}{308} - \frac{19 x y^{2}}{44} + \frac{261 x y z^{3}}{77} - \frac{2031 x y z^{2}}{308} + \frac{15 x y z}{7} + \frac{43 x y}{154} + \frac{47 x z^{3}}{308} - \frac{19 x z^{2}}{44} + \frac{43 x z}{154} + \frac{47 y^{4}}{616} + \frac{31 y^{3} z}{77} - \frac{47 y^{3}}{308} + \frac{111 y^{2} z^{2}}{616} - \frac{3 y^{2} z}{11} + \frac{47 y^{2}}{616} + \frac{31 y z^{3}}{77} - \frac{3 y z^{2}}{11} - \frac{10 y z}{77} + \frac{47 z^{4}}{616} - \frac{47 z^{3}}{308} + \frac{47 z^{2}}{616}$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{21}:\mathbf{V}\mapsto\displaystyle\frac{\sqrt{2}}{2}\int_{e_{0}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle \frac{\sqrt{3}}{3}\\\displaystyle \frac{\sqrt{3}}{3}\\\displaystyle \frac{\sqrt{3}}{3}\end{array}\right)}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)}v$$
where $$e_{0}$$ is the 0th edge.

$$\displaystyle \phi_{21} = \frac{\sqrt{3} \cdot \left(13 x^{4} - 192 x^{3} y z + 26 x^{3} y + 26 x^{3} z - 26 x^{3} - 96 x^{2} y^{2} z - 9 x^{2} y^{2} - 96 x^{2} y z^{2} + 174 x^{2} y z - 20 x^{2} y - 9 x^{2} z^{2} - 20 x^{2} z + 13 x^{2} + 96 x y^{3} z - 2 x y^{3} + 192 x y^{2} z^{2} - 54 x y^{2} z + 8 x y^{2} + 96 x y z^{3} - 54 x y z^{2} - 6 x y - 2 x z^{3} + 8 x z^{2} - 6 x z - y^{4} - 46 y^{3} z + 2 y^{3} - 81 y^{2} z^{2} + 56 y^{2} z - y^{2} - 46 y z^{3} + 56 y z^{2} - 10 y z - z^{4} + 2 z^{3} - z^{2}\right)}{88}$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{22}:\mathbf{V}\mapsto\displaystyle\frac{\sqrt{2}}{2}\int_{e_{0}}\frac{\partial^{2}}{\partial\left(\begin{array}{c}\displaystyle \frac{\sqrt{3}}{3}\\\displaystyle \frac{\sqrt{3}}{3}\\\displaystyle \frac{\sqrt{3}}{3}\end{array}\right)^{2}}v$$
where $$e_{0}$$ is the 0th edge.

$$\displaystyle \phi_{22} = - \frac{31 x^{4}}{308} - \frac{15 x^{3} y z}{77} - \frac{31 x^{3} y}{154} - \frac{31 x^{3} z}{154} + \frac{31 x^{3}}{154} - \frac{156 x^{2} y^{2} z}{77} + \frac{81 x^{2} y^{2}}{616} - \frac{156 x^{2} y z^{2}}{77} + \frac{219 x^{2} y z}{154} + \frac{5 x^{2} y}{44} + \frac{81 x^{2} z^{2}}{616} + \frac{5 x^{2} z}{44} - \frac{31 x^{2}}{308} - \frac{141 x y^{3} z}{77} + \frac{16 x y^{3}}{77} - \frac{282 x y^{2} z^{2}}{77} + \frac{267 x y^{2} z}{77} - \frac{13 x y^{2}}{44} - \frac{141 x y z^{3}}{77} + \frac{267 x y z^{2}}{77} - \frac{11 x y z}{7} + \frac{27 x y}{308} + \frac{16 x z^{3}}{77} - \frac{13 x z^{2}}{44} + \frac{27 x z}{308} + \frac{8 y^{4}}{77} + \frac{295 y^{3} z}{308} - \frac{16 y^{3}}{77} + \frac{513 y^{2} z^{2}}{308} - \frac{29 y^{2} z}{22} + \frac{8 y^{2}}{77} + \frac{295 y z^{3}}{308} - \frac{29 y z^{2}}{22} + \frac{111 y z}{308} + \frac{8 z^{4}}{77} - \frac{16 z^{3}}{77} + \frac{8 z^{2}}{77}$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{23}:\mathbf{V}\mapsto\displaystyle\frac{\sqrt{2}}{2}\int_{e_{1}}\frac{\partial^{2}}{\partial\left(\begin{array}{c}\displaystyle 0\\\displaystyle -1\\\displaystyle 0\end{array}\right)^{2}}v$$
where $$e_{1}$$ is the 1st edge.

$$\displaystyle \phi_{23} = \frac{47 x^{4}}{616} + \frac{261 x^{3} y z}{77} + \frac{47 x^{3} y}{308} + \frac{31 x^{3} z}{77} - \frac{47 x^{3}}{308} + \frac{564 x^{2} y^{2} z}{77} + \frac{129 x^{2} y^{2}}{308} + \frac{522 x^{2} y z^{2}}{77} - \frac{2031 x^{2} y z}{308} - \frac{19 x^{2} y}{44} + \frac{111 x^{2} z^{2}}{616} - \frac{3 x^{2} z}{11} + \frac{47 x^{2}}{616} + \frac{303 x y^{3} z}{77} - \frac{149 x y^{3}}{308} + \frac{564 x y^{2} z^{2}}{77} - \frac{1779 x y^{2} z}{308} + \frac{9 x y^{2}}{44} + \frac{261 x y z^{3}}{77} - \frac{2031 x y z^{2}}{308} + \frac{15 x y z}{7} + \frac{43 x y}{154} + \frac{31 x z^{3}}{77} - \frac{3 x z^{2}}{11} - \frac{10 x z}{77} - \frac{149 y^{4}}{616} - \frac{149 y^{3} z}{308} + \frac{149 y^{3}}{308} + \frac{129 y^{2} z^{2}}{308} + \frac{9 y^{2} z}{44} - \frac{149 y^{2}}{616} + \frac{47 y z^{3}}{308} - \frac{19 y z^{2}}{44} + \frac{43 y z}{154} + \frac{47 z^{4}}{616} - \frac{47 z^{3}}{308} + \frac{47 z^{2}}{616}$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{24}:\mathbf{V}\mapsto\displaystyle\frac{\sqrt{2}}{2}\int_{e_{1}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle \frac{\sqrt{3}}{3}\\\displaystyle \frac{\sqrt{3}}{3}\\\displaystyle \frac{\sqrt{3}}{3}\end{array}\right)}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle 0\\\displaystyle -1\\\displaystyle 0\end{array}\right)}v$$
where $$e_{1}$$ is the 1st edge.

$$\displaystyle \phi_{24} = \frac{\sqrt{3} \left(x^{4} - 96 x^{3} y z + 2 x^{3} y + 46 x^{3} z - 2 x^{3} + 96 x^{2} y^{2} z + 9 x^{2} y^{2} - 192 x^{2} y z^{2} + 54 x^{2} y z - 8 x^{2} y + 81 x^{2} z^{2} - 56 x^{2} z + x^{2} + 192 x y^{3} z - 26 x y^{3} + 96 x y^{2} z^{2} - 174 x y^{2} z + 20 x y^{2} - 96 x y z^{3} + 54 x y z^{2} + 6 x y + 46 x z^{3} - 56 x z^{2} + 10 x z - 13 y^{4} - 26 y^{3} z + 26 y^{3} + 9 y^{2} z^{2} + 20 y^{2} z - 13 y^{2} + 2 y z^{3} - 8 y z^{2} + 6 y z + z^{4} - 2 z^{3} + z^{2}\right)}{88}$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{25}:\mathbf{V}\mapsto\displaystyle\frac{\sqrt{2}}{2}\int_{e_{1}}\frac{\partial^{2}}{\partial\left(\begin{array}{c}\displaystyle \frac{\sqrt{3}}{3}\\\displaystyle \frac{\sqrt{3}}{3}\\\displaystyle \frac{\sqrt{3}}{3}\end{array}\right)^{2}}v$$
where $$e_{1}$$ is the 1st edge.

$$\displaystyle \phi_{25} = \frac{8 x^{4}}{77} - \frac{141 x^{3} y z}{77} + \frac{16 x^{3} y}{77} + \frac{295 x^{3} z}{308} - \frac{16 x^{3}}{77} - \frac{156 x^{2} y^{2} z}{77} + \frac{81 x^{2} y^{2}}{616} - \frac{282 x^{2} y z^{2}}{77} + \frac{267 x^{2} y z}{77} - \frac{13 x^{2} y}{44} + \frac{513 x^{2} z^{2}}{308} - \frac{29 x^{2} z}{22} + \frac{8 x^{2}}{77} - \frac{15 x y^{3} z}{77} - \frac{31 x y^{3}}{154} - \frac{156 x y^{2} z^{2}}{77} + \frac{219 x y^{2} z}{154} + \frac{5 x y^{2}}{44} - \frac{141 x y z^{3}}{77} + \frac{267 x y z^{2}}{77} - \frac{11 x y z}{7} + \frac{27 x y}{308} + \frac{295 x z^{3}}{308} - \frac{29 x z^{2}}{22} + \frac{111 x z}{308} - \frac{31 y^{4}}{308} - \frac{31 y^{3} z}{154} + \frac{31 y^{3}}{154} + \frac{81 y^{2} z^{2}}{616} + \frac{5 y^{2} z}{44} - \frac{31 y^{2}}{308} + \frac{16 y z^{3}}{77} - \frac{13 y z^{2}}{44} + \frac{27 y z}{308} + \frac{8 z^{4}}{77} - \frac{16 z^{3}}{77} + \frac{8 z^{2}}{77}$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{26}:\mathbf{V}\mapsto\displaystyle\frac{\sqrt{2}}{2}\int_{e_{2}}\frac{\partial^{2}}{\partial\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)^{2}}v$$
where $$e_{2}$$ is the 2nd edge.

$$\displaystyle \phi_{26} = \frac{47 x^{4}}{616} + \frac{261 x^{3} y z}{77} + \frac{31 x^{3} y}{77} + \frac{47 x^{3} z}{308} - \frac{47 x^{3}}{308} + \frac{522 x^{2} y^{2} z}{77} + \frac{111 x^{2} y^{2}}{616} + \frac{564 x^{2} y z^{2}}{77} - \frac{2031 x^{2} y z}{308} - \frac{3 x^{2} y}{11} + \frac{129 x^{2} z^{2}}{308} - \frac{19 x^{2} z}{44} + \frac{47 x^{2}}{616} + \frac{261 x y^{3} z}{77} + \frac{31 x y^{3}}{77} + \frac{564 x y^{2} z^{2}}{77} - \frac{2031 x y^{2} z}{308} - \frac{3 x y^{2}}{11} + \frac{303 x y z^{3}}{77} - \frac{1779 x y z^{2}}{308} + \frac{15 x y z}{7} - \frac{10 x y}{77} - \frac{149 x z^{3}}{308} + \frac{9 x z^{2}}{44} + \frac{43 x z}{154} + \frac{47 y^{4}}{616} + \frac{47 y^{3} z}{308} - \frac{47 y^{3}}{308} + \frac{129 y^{2} z^{2}}{308} - \frac{19 y^{2} z}{44} + \frac{47 y^{2}}{616} - \frac{149 y z^{3}}{308} + \frac{9 y z^{2}}{44} + \frac{43 y z}{154} - \frac{149 z^{4}}{616} + \frac{149 z^{3}}{308} - \frac{149 z^{2}}{616}$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{27}:\mathbf{V}\mapsto\displaystyle\frac{\sqrt{2}}{2}\int_{e_{2}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle \frac{\sqrt{3}}{3}\\\displaystyle \frac{\sqrt{3}}{3}\\\displaystyle \frac{\sqrt{3}}{3}\end{array}\right)}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)}v$$
where $$e_{2}$$ is the 2nd edge.

$$\displaystyle \phi_{27} = \frac{\sqrt{3} \left(- x^{4} + 96 x^{3} y z - 46 x^{3} y - 2 x^{3} z + 2 x^{3} + 192 x^{2} y^{2} z - 81 x^{2} y^{2} - 96 x^{2} y z^{2} - 54 x^{2} y z + 56 x^{2} y - 9 x^{2} z^{2} + 8 x^{2} z - x^{2} + 96 x y^{3} z - 46 x y^{3} - 96 x y^{2} z^{2} - 54 x y^{2} z + 56 x y^{2} - 192 x y z^{3} + 174 x y z^{2} - 10 x y + 26 x z^{3} - 20 x z^{2} - 6 x z - y^{4} - 2 y^{3} z + 2 y^{3} - 9 y^{2} z^{2} + 8 y^{2} z - y^{2} + 26 y z^{3} - 20 y z^{2} - 6 y z + 13 z^{4} - 26 z^{3} + 13 z^{2}\right)}{88}$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{28}:\mathbf{V}\mapsto\displaystyle\frac{\sqrt{2}}{2}\int_{e_{2}}\frac{\partial^{2}}{\partial\left(\begin{array}{c}\displaystyle \frac{\sqrt{3}}{3}\\\displaystyle \frac{\sqrt{3}}{3}\\\displaystyle \frac{\sqrt{3}}{3}\end{array}\right)^{2}}v$$
where $$e_{2}$$ is the 2nd edge.

$$\displaystyle \phi_{28} = \frac{8 x^{4}}{77} - \frac{141 x^{3} y z}{77} + \frac{295 x^{3} y}{308} + \frac{16 x^{3} z}{77} - \frac{16 x^{3}}{77} - \frac{282 x^{2} y^{2} z}{77} + \frac{513 x^{2} y^{2}}{308} - \frac{156 x^{2} y z^{2}}{77} + \frac{267 x^{2} y z}{77} - \frac{29 x^{2} y}{22} + \frac{81 x^{2} z^{2}}{616} - \frac{13 x^{2} z}{44} + \frac{8 x^{2}}{77} - \frac{141 x y^{3} z}{77} + \frac{295 x y^{3}}{308} - \frac{156 x y^{2} z^{2}}{77} + \frac{267 x y^{2} z}{77} - \frac{29 x y^{2}}{22} - \frac{15 x y z^{3}}{77} + \frac{219 x y z^{2}}{154} - \frac{11 x y z}{7} + \frac{111 x y}{308} - \frac{31 x z^{3}}{154} + \frac{5 x z^{2}}{44} + \frac{27 x z}{308} + \frac{8 y^{4}}{77} + \frac{16 y^{3} z}{77} - \frac{16 y^{3}}{77} + \frac{81 y^{2} z^{2}}{616} - \frac{13 y^{2} z}{44} + \frac{8 y^{2}}{77} - \frac{31 y z^{3}}{154} + \frac{5 y z^{2}}{44} + \frac{27 y z}{308} - \frac{31 z^{4}}{308} + \frac{31 z^{3}}{154} - \frac{31 z^{2}}{308}$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{29}:\mathbf{V}\mapsto\displaystyle\int_{e_{3}}\frac{\partial^{2}}{\partial\left(\begin{array}{c}\displaystyle 0\\\displaystyle -1\\\displaystyle 0\end{array}\right)^{2}}v$$
where $$e_{3}$$ is the 3th edge.

$$\displaystyle \phi_{29} = \frac{17 x^{4}}{154} - \frac{222 x^{3} y z}{77} + \frac{17 x^{3} y}{77} + \frac{17 x^{3} z}{77} - \frac{17 x^{3}}{77} - \frac{276 x^{2} y^{2} z}{77} + \frac{45 x^{2} y^{2}}{308} - \frac{276 x^{2} y z^{2}}{77} + \frac{327 x^{2} y z}{77} - \frac{7 x^{2} y}{22} + \frac{45 x^{2} z^{2}}{308} - \frac{7 x^{2} z}{22} + \frac{17 x^{2}}{154} - \frac{54 x y^{3} z}{77} + \frac{52 x y^{3}}{77} - \frac{108 x y^{2} z^{2}}{77} + \frac{117 x y^{2} z}{77} - \frac{17 x y^{2}}{22} - \frac{54 x y z^{3}}{77} + \frac{117 x y z^{2}}{77} - \frac{6 x y z}{7} + \frac{15 x y}{154} - \frac{25 x z^{3}}{77} + \frac{5 x z^{2}}{22} + \frac{15 x z}{154} + \frac{26 y^{4}}{77} - \frac{25 y^{3} z}{77} - \frac{52 y^{3}}{77} - \frac{123 y^{2} z^{2}}{308} + \frac{13 y^{2} z}{22} + \frac{26 y^{2}}{77} - \frac{25 y z^{3}}{77} + \frac{13 y z^{2}}{22} - \frac{41 y z}{154} - \frac{25 z^{4}}{154} + \frac{25 z^{3}}{77} - \frac{25 z^{2}}{154}$$

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{30}:\mathbf{V}\mapsto\displaystyle\int_{e_{3}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle 0\\\displaystyle -1\\\displaystyle 0\end{array}\right)}v$$
where $$e_{3}$$ is the 3th edge.

$$\displaystyle \phi_{30} = \frac{23 x^{4}}{77} - \frac{492 x^{3} y z}{77} + \frac{46 x^{3} y}{77} + \frac{46 x^{3} z}{77} - \frac{46 x^{3}}{77} - \frac{984 x^{2} y^{2} z}{77} + \frac{15 x^{2} y^{2}}{77} - \frac{564 x^{2} y z^{2}}{77} + \frac{912 x^{2} y z}{77} - \frac{8 x^{2} y}{11} + \frac{51 x^{2} z^{2}}{154} - \frac{9 x^{2} z}{11} + \frac{23 x^{2}}{77} - \frac{492 x y^{3} z}{77} + \frac{46 x y^{3}}{77} - \frac{564 x y^{2} z^{2}}{77} + \frac{912 x y^{2} z}{77} - \frac{8 x y^{2}}{11} - \frac{72 x y z^{3}}{77} + \frac{387 x y z^{2}}{77} - \frac{36 x y z}{7} + \frac{10 x y}{77} - \frac{59 x z^{3}}{77} + \frac{6 x z^{2}}{11} + \frac{17 x z}{77} + \frac{23 y^{4}}{77} + \frac{46 y^{3} z}{77} - \frac{46 y^{3}}{77} + \frac{51 y^{2} z^{2}}{154} - \frac{9 y^{2} z}{11} + \frac{23 y^{2}}{77} - \frac{59 y z^{3}}{77} + \frac{6 y z^{2}}{11} + \frac{17 y z}{77} - \frac{59 z^{4}}{154} + \frac{59 z^{3}}{77} - \frac{59 z^{2}}{154}$$

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{31}:\mathbf{V}\mapsto\displaystyle\int_{e_{3}}\frac{\partial^{2}}{\partial\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)^{2}}v$$
where $$e_{3}$$ is the 3th edge.

$$\displaystyle \phi_{31} = \frac{26 x^{4}}{77} - \frac{54 x^{3} y z}{77} + \frac{52 x^{3} y}{77} - \frac{25 x^{3} z}{77} - \frac{52 x^{3}}{77} - \frac{276 x^{2} y^{2} z}{77} + \frac{45 x^{2} y^{2}}{308} - \frac{108 x^{2} y z^{2}}{77} + \frac{117 x^{2} y z}{77} - \frac{17 x^{2} y}{22} - \frac{123 x^{2} z^{2}}{308} + \frac{13 x^{2} z}{22} + \frac{26 x^{2}}{77} - \frac{222 x y^{3} z}{77} + \frac{17 x y^{3}}{77} - \frac{276 x y^{2} z^{2}}{77} + \frac{327 x y^{2} z}{77} - \frac{7 x y^{2}}{22} - \frac{54 x y z^{3}}{77} + \frac{117 x y z^{2}}{77} - \frac{6 x y z}{7} + \frac{15 x y}{154} - \frac{25 x z^{3}}{77} + \frac{13 x z^{2}}{22} - \frac{41 x z}{154} + \frac{17 y^{4}}{154} + \frac{17 y^{3} z}{77} - \frac{17 y^{3}}{77} + \frac{45 y^{2} z^{2}}{308} - \frac{7 y^{2} z}{22} + \frac{17 y^{2}}{154} - \frac{25 y z^{3}}{77} + \frac{5 y z^{2}}{22} + \frac{15 y z}{154} - \frac{25 z^{4}}{154} + \frac{25 z^{3}}{77} - \frac{25 z^{2}}{154}$$

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{32}:\mathbf{V}\mapsto\displaystyle\int_{e_{4}}\frac{\partial^{2}}{\partial\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)^{2}}v$$
where $$e_{4}$$ is the 4th edge.

$$\displaystyle \phi_{32} = \frac{17 x^{4}}{154} - \frac{222 x^{3} y z}{77} + \frac{17 x^{3} y}{77} + \frac{17 x^{3} z}{77} - \frac{17 x^{3}}{77} - \frac{276 x^{2} y^{2} z}{77} + \frac{45 x^{2} y^{2}}{308} - \frac{276 x^{2} y z^{2}}{77} + \frac{327 x^{2} y z}{77} - \frac{7 x^{2} y}{22} + \frac{45 x^{2} z^{2}}{308} - \frac{7 x^{2} z}{22} + \frac{17 x^{2}}{154} - \frac{54 x y^{3} z}{77} - \frac{25 x y^{3}}{77} - \frac{108 x y^{2} z^{2}}{77} + \frac{117 x y^{2} z}{77} + \frac{5 x y^{2}}{22} - \frac{54 x y z^{3}}{77} + \frac{117 x y z^{2}}{77} - \frac{6 x y z}{7} + \frac{15 x y}{154} + \frac{52 x z^{3}}{77} - \frac{17 x z^{2}}{22} + \frac{15 x z}{154} - \frac{25 y^{4}}{154} - \frac{25 y^{3} z}{77} + \frac{25 y^{3}}{77} - \frac{123 y^{2} z^{2}}{308} + \frac{13 y^{2} z}{22} - \frac{25 y^{2}}{154} - \frac{25 y z^{3}}{77} + \frac{13 y z^{2}}{22} - \frac{41 y z}{154} + \frac{26 z^{4}}{77} - \frac{52 z^{3}}{77} + \frac{26 z^{2}}{77}$$

This DOF is associated with edge 4 of the reference element.
$$\displaystyle l_{33}:\mathbf{V}\mapsto\displaystyle\int_{e_{4}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)}v$$
where $$e_{4}$$ is the 4th edge.

$$\displaystyle \phi_{33} = - \frac{23 x^{4}}{77} + \frac{492 x^{3} y z}{77} - \frac{46 x^{3} y}{77} - \frac{46 x^{3} z}{77} + \frac{46 x^{3}}{77} + \frac{564 x^{2} y^{2} z}{77} - \frac{51 x^{2} y^{2}}{154} + \frac{984 x^{2} y z^{2}}{77} - \frac{912 x^{2} y z}{77} + \frac{9 x^{2} y}{11} - \frac{15 x^{2} z^{2}}{77} + \frac{8 x^{2} z}{11} - \frac{23 x^{2}}{77} + \frac{72 x y^{3} z}{77} + \frac{59 x y^{3}}{77} + \frac{564 x y^{2} z^{2}}{77} - \frac{387 x y^{2} z}{77} - \frac{6 x y^{2}}{11} + \frac{492 x y z^{3}}{77} - \frac{912 x y z^{2}}{77} + \frac{36 x y z}{7} - \frac{17 x y}{77} - \frac{46 x z^{3}}{77} + \frac{8 x z^{2}}{11} - \frac{10 x z}{77} + \frac{59 y^{4}}{154} + \frac{59 y^{3} z}{77} - \frac{59 y^{3}}{77} - \frac{51 y^{2} z^{2}}{154} - \frac{6 y^{2} z}{11} + \frac{59 y^{2}}{154} - \frac{46 y z^{3}}{77} + \frac{9 y z^{2}}{11} - \frac{17 y z}{77} - \frac{23 z^{4}}{77} + \frac{46 z^{3}}{77} - \frac{23 z^{2}}{77}$$

This DOF is associated with edge 4 of the reference element.
$$\displaystyle l_{34}:\mathbf{V}\mapsto\displaystyle\int_{e_{4}}\frac{\partial^{2}}{\partial\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)^{2}}v$$
where $$e_{4}$$ is the 4th edge.

$$\displaystyle \phi_{34} = \frac{26 x^{4}}{77} - \frac{54 x^{3} y z}{77} - \frac{25 x^{3} y}{77} + \frac{52 x^{3} z}{77} - \frac{52 x^{3}}{77} - \frac{108 x^{2} y^{2} z}{77} - \frac{123 x^{2} y^{2}}{308} - \frac{276 x^{2} y z^{2}}{77} + \frac{117 x^{2} y z}{77} + \frac{13 x^{2} y}{22} + \frac{45 x^{2} z^{2}}{308} - \frac{17 x^{2} z}{22} + \frac{26 x^{2}}{77} - \frac{54 x y^{3} z}{77} - \frac{25 x y^{3}}{77} - \frac{276 x y^{2} z^{2}}{77} + \frac{117 x y^{2} z}{77} + \frac{13 x y^{2}}{22} - \frac{222 x y z^{3}}{77} + \frac{327 x y z^{2}}{77} - \frac{6 x y z}{7} - \frac{41 x y}{154} + \frac{17 x z^{3}}{77} - \frac{7 x z^{2}}{22} + \frac{15 x z}{154} - \frac{25 y^{4}}{154} - \frac{25 y^{3} z}{77} + \frac{25 y^{3}}{77} + \frac{45 y^{2} z^{2}}{308} + \frac{5 y^{2} z}{22} - \frac{25 y^{2}}{154} + \frac{17 y z^{3}}{77} - \frac{7 y z^{2}}{22} + \frac{15 y z}{154} + \frac{17 z^{4}}{154} - \frac{17 z^{3}}{77} + \frac{17 z^{2}}{154}$$

This DOF is associated with edge 4 of the reference element.
$$\displaystyle l_{35}:\mathbf{V}\mapsto\displaystyle\int_{e_{5}}\frac{\partial^{2}}{\partial\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)^{2}}v$$
where $$e_{5}$$ is the 5th edge.

$$\displaystyle \phi_{35} = - \frac{25 x^{4}}{154} - \frac{54 x^{3} y z}{77} - \frac{25 x^{3} y}{77} - \frac{25 x^{3} z}{77} + \frac{25 x^{3}}{77} - \frac{276 x^{2} y^{2} z}{77} + \frac{45 x^{2} y^{2}}{308} - \frac{108 x^{2} y z^{2}}{77} + \frac{117 x^{2} y z}{77} + \frac{5 x^{2} y}{22} - \frac{123 x^{2} z^{2}}{308} + \frac{13 x^{2} z}{22} - \frac{25 x^{2}}{154} - \frac{222 x y^{3} z}{77} + \frac{17 x y^{3}}{77} - \frac{276 x y^{2} z^{2}}{77} + \frac{327 x y^{2} z}{77} - \frac{7 x y^{2}}{22} - \frac{54 x y z^{3}}{77} + \frac{117 x y z^{2}}{77} - \frac{6 x y z}{7} + \frac{15 x y}{154} - \frac{25 x z^{3}}{77} + \frac{13 x z^{2}}{22} - \frac{41 x z}{154} + \frac{17 y^{4}}{154} + \frac{17 y^{3} z}{77} - \frac{17 y^{3}}{77} + \frac{45 y^{2} z^{2}}{308} - \frac{7 y^{2} z}{22} + \frac{17 y^{2}}{154} + \frac{52 y z^{3}}{77} - \frac{17 y z^{2}}{22} + \frac{15 y z}{154} + \frac{26 z^{4}}{77} - \frac{52 z^{3}}{77} + \frac{26 z^{2}}{77}$$

This DOF is associated with edge 5 of the reference element.
$$\displaystyle l_{36}:\mathbf{V}\mapsto\displaystyle\int_{e_{5}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle 0\\\displaystyle -1\\\displaystyle 0\end{array}\right)}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)}v$$
where $$e_{5}$$ is the 5th edge.

$$\displaystyle \phi_{36} = - \frac{59 x^{4}}{154} - \frac{72 x^{3} y z}{77} - \frac{59 x^{3} y}{77} - \frac{59 x^{3} z}{77} + \frac{59 x^{3}}{77} - \frac{564 x^{2} y^{2} z}{77} + \frac{51 x^{2} y^{2}}{154} - \frac{564 x^{2} y z^{2}}{77} + \frac{387 x^{2} y z}{77} + \frac{6 x^{2} y}{11} + \frac{51 x^{2} z^{2}}{154} + \frac{6 x^{2} z}{11} - \frac{59 x^{2}}{154} - \frac{492 x y^{3} z}{77} + \frac{46 x y^{3}}{77} - \frac{984 x y^{2} z^{2}}{77} + \frac{912 x y^{2} z}{77} - \frac{9 x y^{2}}{11} - \frac{492 x y z^{3}}{77} + \frac{912 x y z^{2}}{77} - \frac{36 x y z}{7} + \frac{17 x y}{77} + \frac{46 x z^{3}}{77} - \frac{9 x z^{2}}{11} + \frac{17 x z}{77} + \frac{23 y^{4}}{77} + \frac{46 y^{3} z}{77} - \frac{46 y^{3}}{77} + \frac{15 y^{2} z^{2}}{77} - \frac{8 y^{2} z}{11} + \frac{23 y^{2}}{77} + \frac{46 y z^{3}}{77} - \frac{8 y z^{2}}{11} + \frac{10 y z}{77} + \frac{23 z^{4}}{77} - \frac{46 z^{3}}{77} + \frac{23 z^{2}}{77}$$

This DOF is associated with edge 5 of the reference element.
$$\displaystyle l_{37}:\mathbf{V}\mapsto\displaystyle\int_{e_{5}}\frac{\partial^{2}}{\partial\left(\begin{array}{c}\displaystyle 0\\\displaystyle -1\\\displaystyle 0\end{array}\right)^{2}}v$$
where $$e_{5}$$ is the 5th edge.

$$\displaystyle \phi_{37} = - \frac{25 x^{4}}{154} - \frac{54 x^{3} y z}{77} - \frac{25 x^{3} y}{77} - \frac{25 x^{3} z}{77} + \frac{25 x^{3}}{77} - \frac{108 x^{2} y^{2} z}{77} - \frac{123 x^{2} y^{2}}{308} - \frac{276 x^{2} y z^{2}}{77} + \frac{117 x^{2} y z}{77} + \frac{13 x^{2} y}{22} + \frac{45 x^{2} z^{2}}{308} + \frac{5 x^{2} z}{22} - \frac{25 x^{2}}{154} - \frac{54 x y^{3} z}{77} - \frac{25 x y^{3}}{77} - \frac{276 x y^{2} z^{2}}{77} + \frac{117 x y^{2} z}{77} + \frac{13 x y^{2}}{22} - \frac{222 x y z^{3}}{77} + \frac{327 x y z^{2}}{77} - \frac{6 x y z}{7} - \frac{41 x y}{154} + \frac{17 x z^{3}}{77} - \frac{7 x z^{2}}{22} + \frac{15 x z}{154} + \frac{26 y^{4}}{77} + \frac{52 y^{3} z}{77} - \frac{52 y^{3}}{77} + \frac{45 y^{2} z^{2}}{308} - \frac{17 y^{2} z}{22} + \frac{26 y^{2}}{77} + \frac{17 y z^{3}}{77} - \frac{7 y z^{2}}{22} + \frac{15 y z}{154} + \frac{17 z^{4}}{154} - \frac{17 z^{3}}{77} + \frac{17 z^{2}}{154}$$

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

## References

• Wu, Shuonan and Xu, Jinchao. Nonconforming finite element spaces for 2mth order partial differential equations on Rn simplical grids when m=n+1, Mathematics of computation 88, 531–551, 2019. [DOI: 10.1090/mcom/3361] [BibTeX]

## DefElement stats

 Element added 08 June 2021 Element last updated 10 February 2022