Degree 3 Morley–Wang–Xu on a tetrahedron

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In this example:

$R$ is the reference tetrahedron. The following numbering of the subentities of the reference is used:

$V$ is spanned by: $1$ , $x$ , $x^{2}$ , $x^{3}$ , $y$ , $x y$ , $x^{2} y$ , $y^{2}$ , $x y^{2}$ , $y^{3}$ , $z$ , $x z$ , $x^{2} z$ , $y z$ , $x y z$ , $y^{2} z$ , $z^{2}$ , $x z^{2}$ , $y z^{2}$ , $z^{3}$
$L = {l_{0}, . . ., l_{19}}$
Functionals and basis functions:

l_{0} : v \mapsto v (0, 0, 0)

ϕ_{0} = - 6 x y z + 2 x y + 2 x z - x + 2 y z - y - z + 1

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

l_{1} : v \mapsto v (1, 0, 0)

ϕ_{1} = \frac{x^{3}}{3} - \frac{x^{2} y}{2} - \frac{x^{2} z}{2} + \frac{x^{2}}{3} - \frac{x y^{2}}{2} + 2 x y z - \frac{x y}{3} - \frac{x z^{2}}{2} - \frac{x z}{3} + \frac{x}{3} - \frac{y^{3}}{6} + y^{2} z - \frac{y^{2}}{6} + y z^{2} - \frac{4 y z}{3} + \frac{y}{3} - \frac{z^{3}}{6} - \frac{z^{2}}{6} + \frac{z}{3}

This DOF is associated with vertex 1 of the reference element.

l_{2} : v \mapsto v (0, 1, 0)

ϕ_{2} = - \frac{x^{3}}{6} - \frac{x^{2} y}{2} + x^{2} z - \frac{x^{2}}{6} - \frac{x y^{2}}{2} + 2 x y z - \frac{x y}{3} + x z^{2} - \frac{4 x z}{3} + \frac{x}{3} + \frac{y^{3}}{3} - \frac{y^{2} z}{2} + \frac{y^{2}}{3} - \frac{y z^{2}}{2} - \frac{y z}{3} + \frac{y}{3} - \frac{z^{3}}{6} - \frac{z^{2}}{6} + \frac{z}{3}

This DOF is associated with vertex 2 of the reference element.

l_{3} : v \mapsto v (0, 0, 1)

ϕ_{3} = - \frac{x^{3}}{6} + x^{2} y - \frac{x^{2} z}{2} - \frac{x^{2}}{6} + x y^{2} + 2 x y z - \frac{4 x y}{3} - \frac{x z^{2}}{2} - \frac{x z}{3} + \frac{x}{3} - \frac{y^{3}}{6} - \frac{y^{2} z}{2} - \frac{y^{2}}{6} - \frac{y z^{2}}{2} - \frac{y z}{3} + \frac{y}{3} + \frac{z^{3}}{3} + \frac{z^{2}}{3} + \frac{z}{3}

This DOF is associated with vertex 3 of the reference element.

l_{4} : V \mapsto \frac{\sqrt{2}}{2} \int_{e_{0}} \frac{\partial}{\partial (\begin{matrix} \frac{\sqrt{3}}{3} \\ \frac{\sqrt{3}}{3} \\ \frac{\sqrt{3}}{3} \end{matrix})} v

where

e_{0}

is the 0th edge.

ϕ_{4} = \frac{\sqrt{3} (- 4 x^{3} - 12 x^{2} y - 12 x^{2} z + 8 x^{2} - 12 x y^{2} - 24 x y z + 16 x y - 12 x z^{2} + 16 x z - 4 x + 5 y^{3} + 15 y^{2} z - y^{2} + 15 y z^{2} - 2 y z - 4 y + 5 z^{3} - z^{2} - 4 z)}{18}

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

l_{5} : V \mapsto \frac{\sqrt{2}}{2} \int_{e_{0}} \frac{\partial}{\partial (\begin{matrix} 1 \\ 0 \\ 0 \end{matrix})} v

where

e_{0}

is the 0th edge.

ϕ_{5} = - \frac{2 x^{3}}{3} - 2 x^{2} y - 2 x^{2} z + \frac{4 x^{2}}{3} + x y^{2} + 2 x y z + \frac{2 x y}{3} + x z^{2} + \frac{2 x z}{3} - \frac{2 x}{3} - \frac{y^{3}}{6} - \frac{y^{2} z}{2} - \frac{y^{2}}{6} - \frac{y z^{2}}{2} - \frac{y z}{3} + \frac{y}{3} - \frac{z^{3}}{6} - \frac{z^{2}}{6} + \frac{z}{3}

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

l_{6} : V \mapsto \frac{\sqrt{2}}{2} \int_{e_{1}} \frac{\partial}{\partial (\begin{matrix} \frac{\sqrt{3}}{3} \\ \frac{\sqrt{3}}{3} \\ \frac{\sqrt{3}}{3} \end{matrix})} v

where

e_{1}

is the 1st edge.

ϕ_{6} = \frac{\sqrt{3} (5 x^{3} - 12 x^{2} y + 15 x^{2} z - x^{2} - 12 x y^{2} - 24 x y z + 16 x y + 15 x z^{2} - 2 x z - 4 x - 4 y^{3} - 12 y^{2} z + 8 y^{2} - 12 y z^{2} + 16 y z - 4 y + 5 z^{3} - z^{2} - 4 z)}{18}

This DOF is associated with edge 1 of the reference element.

l_{7} : V \mapsto \frac{\sqrt{2}}{2} \int_{e_{1}} \frac{\partial}{\partial (\begin{matrix} 0 \\ - 1 \\ 0 \end{matrix})} v

where

e_{1}

is the 1st edge.

ϕ_{7} = \frac{x^{3}}{6} - x^{2} y + \frac{x^{2} z}{2} + \frac{x^{2}}{6} + 2 x y^{2} - 2 x y z - \frac{2 x y}{3} + \frac{x z^{2}}{2} + \frac{x z}{3} - \frac{x}{3} + \frac{2 y^{3}}{3} + 2 y^{2} z - \frac{4 y^{2}}{3} - y z^{2} - \frac{2 y z}{3} + \frac{2 y}{3} + \frac{z^{3}}{6} + \frac{z^{2}}{6} - \frac{z}{3}

This DOF is associated with edge 1 of the reference element.

l_{8} : V \mapsto \frac{\sqrt{2}}{2} \int_{e_{2}} \frac{\partial}{\partial (\begin{matrix} \frac{\sqrt{3}}{3} \\ \frac{\sqrt{3}}{3} \\ \frac{\sqrt{3}}{3} \end{matrix})} v

where

e_{2}

is the 2nd edge.

ϕ_{8} = \frac{\sqrt{3} (5 x^{3} + 15 x^{2} y - 12 x^{2} z - x^{2} + 15 x y^{2} - 24 x y z - 2 x y - 12 x z^{2} + 16 x z - 4 x + 5 y^{3} - 12 y^{2} z - y^{2} - 12 y z^{2} + 16 y z - 4 y - 4 z^{3} + 8 z^{2} - 4 z)}{18}

This DOF is associated with edge 2 of the reference element.

l_{9} : V \mapsto \frac{\sqrt{2}}{2} \int_{e_{2}} \frac{\partial}{\partial (\begin{matrix} 0 \\ 0 \\ 1 \end{matrix})} v

where

e_{2}

is the 2nd edge.

ϕ_{9} = - \frac{x^{3}}{6} - \frac{x^{2} y}{2} + x^{2} z - \frac{x^{2}}{6} - \frac{x y^{2}}{2} + 2 x y z - \frac{x y}{3} - 2 x z^{2} + \frac{2 x z}{3} + \frac{x}{3} - \frac{y^{3}}{6} + y^{2} z - \frac{y^{2}}{6} - 2 y z^{2} + \frac{2 y z}{3} + \frac{y}{3} - \frac{2 z^{3}}{3} + \frac{4 z^{2}}{3} - \frac{2 z}{3}

This DOF is associated with edge 2 of the reference element.

l_{10} : V \mapsto \int_{e_{3}} \frac{\partial}{\partial (\begin{matrix} 1 \\ 0 \\ 0 \end{matrix})} v

where

e_{3}

is the 3rd edge.

ϕ_{10} = x (3 x y - x - 2 y + 1)

This DOF is associated with edge 3 of the reference element.

l_{11} : V \mapsto \int_{e_{3}} \frac{\partial}{\partial (\begin{matrix} 0 \\ - 1 \\ 0 \end{matrix})} v

where

e_{3}

is the 3rd edge.

ϕ_{11} = y (- 3 x y + 2 x + y - 1)

This DOF is associated with edge 3 of the reference element.

l_{12} : V \mapsto \int_{e_{4}} \frac{\partial}{\partial (\begin{matrix} 1 \\ 0 \\ 0 \end{matrix})} v

where

e_{4}

is the 4th edge.

ϕ_{12} = x (3 x z - x - 2 z + 1)

This DOF is associated with edge 4 of the reference element.

l_{13} : V \mapsto \int_{e_{4}} \frac{\partial}{\partial (\begin{matrix} 0 \\ 0 \\ 1 \end{matrix})} v

where

e_{4}

is the 4th edge.

ϕ_{13} = z (3 x z - 2 x - z + 1)

This DOF is associated with edge 4 of the reference element.

l_{14} : V \mapsto \int_{e_{5}} \frac{\partial}{\partial (\begin{matrix} 0 \\ - 1 \\ 0 \end{matrix})} v

where

e_{5}

is the 5th edge.

ϕ_{14} = y (- 3 y z + y + 2 z - 1)

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

l_{15} : V \mapsto \int_{e_{5}} \frac{\partial}{\partial (\begin{matrix} 0 \\ 0 \\ 1 \end{matrix})} v

where

e_{5}

is the 5th edge.

ϕ_{15} = z (3 y z - 2 y - z + 1)

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

l_{16} : V \mapsto \frac{\sqrt{3}}{3} \int_{f_{0}} \frac{\partial^{2}}{\partial {(\begin{matrix} \frac{\sqrt{3}}{3} \\ \frac{\sqrt{3}}{3} \\ \frac{\sqrt{3}}{3} \end{matrix})}^{2}} v

where

f_{0}

is the 0th face.

ϕ_{16} = \frac{x^{3}}{3} + x^{2} y + x^{2} z - \frac{2 x^{2}}{3} + x y^{2} + 2 x y z - \frac{4 x y}{3} + x z^{2} - \frac{4 x z}{3} + \frac{x}{3} + \frac{y^{3}}{3} + y^{2} z - \frac{2 y^{2}}{3} + y z^{2} - \frac{4 y z}{3} + \frac{y}{3} + \frac{z^{3}}{3} - \frac{2 z^{2}}{3} + \frac{z}{3}

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

l_{17} : V \mapsto \int_{f_{1}} \frac{\partial^{2}}{\partial {(\begin{matrix} 1 \\ 0 \\ 0 \end{matrix})}^{2}} v

where

f_{1}

is the 1st face.

ϕ_{17} = x^{2} (1 - x)

This DOF is associated with face 1 of the reference element.

l_{18} : V \mapsto \int_{f_{2}} \frac{\partial^{2}}{\partial {(\begin{matrix} 0 \\ - 1 \\ 0 \end{matrix})}^{2}} v

where

f_{2}

is the 2nd face.

ϕ_{18} = y^{2} (1 - y)

This DOF is associated with face 2 of the reference element.

l_{19} : V \mapsto \int_{f_{3}} \frac{\partial^{2}}{\partial {(\begin{matrix} 0 \\ 0 \\ 1 \end{matrix})}^{2}} v

where

f_{3}

is the 3rd face.

ϕ_{19} = z^{2} (1 - z)

This DOF is associated with face 3 of the reference element.