Degree 2 Morley–Wang–Xu on a tetrahedron

In this example:

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

$V$ is spanned by: $1$ , $x$ , $x^{2}$ , $y$ , $x y$ , $y^{2}$ , $z$ , $x z$ , $y z$ , $z^{2}$
$L = {l_{0}, . . ., l_{9}}$
Functionals and basis functions:

l_{0} : v \mapsto \int_{e_{0}} (\frac{\sqrt{2}}{2}) v

where

e_{0}

is the 0th edge.

ϕ_{0} = - 2 x^{2} - 4 x y - 4 x z + \frac{8 x}{3} + y^{2} + 2 y z + \frac{2 y}{3} + z^{2} + \frac{2 z}{3} - \frac{2}{3}

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

l_{1} : v \mapsto \int_{e_{1}} (\frac{\sqrt{2}}{2}) v

where

e_{1}

is the 1st edge.

ϕ_{1} = x^{2} - 4 x y + 2 x z + \frac{2 x}{3} - 2 y^{2} - 4 y z + \frac{8 y}{3} + z^{2} + \frac{2 z}{3} - \frac{2}{3}

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

l_{2} : v \mapsto \int_{e_{2}} (\frac{\sqrt{2}}{2}) v

where

e_{2}

is the 2nd edge.

ϕ_{2} = x^{2} + 2 x y - 4 x z + \frac{2 x}{3} + y^{2} - 4 y z + \frac{2 y}{3} - 2 z^{2} + \frac{8 z}{3} - \frac{2}{3}

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

l_{3} : v \mapsto \int_{e_{3}} v

where

e_{3}

is the 3rd edge.

ϕ_{3} = 6 x y - 2 x - 2 y + 1

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

l_{4} : v \mapsto \int_{e_{4}} v

where

e_{4}

is the 4th edge.

ϕ_{4} = 6 x z - 2 x - 2 z + 1

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

l_{5} : v \mapsto \int_{e_{5}} v

where

e_{5}

is the 5th edge.

ϕ_{5} = 6 y z - 2 y - 2 z + 1

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

l_{6} : V \mapsto \frac{\sqrt{3}}{3} \int_{f_{0}} \frac{\partial}{\partial (\begin{matrix} \frac{\sqrt{3}}{3} \\ \frac{\sqrt{3}}{3} \\ \frac{\sqrt{3}}{3} \end{matrix})} v

where

f_{0}

is the 0th face.

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

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

l_{7} : V \mapsto \int_{f_{1}} \frac{\partial}{\partial (\begin{matrix} 1 \\ 0 \\ 0 \end{matrix})} v

where

f_{1}

is the 1st face.

ϕ_{7} = x (2 - 3 x)

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

l_{8} : V \mapsto \int_{f_{2}} \frac{\partial}{\partial (\begin{matrix} 0 \\ - 1 \\ 0 \end{matrix})} v

where

f_{2}

is the 2nd face.

ϕ_{8} = y (3 y - 2)

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

l_{9} : V \mapsto \int_{f_{3}} \frac{\partial}{\partial (\begin{matrix} 0 \\ 0 \\ 1 \end{matrix})} v

where

f_{3}

is the 3rd face.

ϕ_{9} = z (2 - 3 z)

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