an encyclopedia of finite element definitions

# Morley–Wang–Xu

 Orders $$1\leqslant k\leqslant {'interval': 1, 'triangle': 2, 'tetrahedron': 3}$$ Reference elements interval, triangle, tetrahedron Polynomial set $$\mathcal{P}_{k}$$↓ 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: $$k+1$$ (A000027)triangle: $$(k+1)(k+2)/2$$ (A000217)tetrahedron: $$(k+1)(k+2)(k+3)/6$$ (A000292) Categories Scalar-valued elements

## Implementations

 Symfem string "MWX"↓ Show Symfem examples ↓

## Examples

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

$$\displaystyle \phi_{0} = 1 - x$$

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

$$\displaystyle \phi_{1} = x$$

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: $$1$$, $$x$$, $$y$$
• $$\mathcal{L}=\{l_0,...,l_{2}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\mathbf{V}\mapsto\displaystyle\int_{e_{0}}(\tfrac{\sqrt{2}}{2})v$$

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

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{1}:\mathbf{V}\mapsto\displaystyle\int_{e_{1}}v$$

$$\displaystyle \phi_{1} = 1 - 2 x$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{2}:\mathbf{V}\mapsto\displaystyle\int_{e_{2}}v$$

$$\displaystyle \phi_{2} = 1 - 2 y$$

This DOF is associated with edge 2 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: $$1$$, $$x$$, $$x^{2}$$, $$y$$, $$x y$$, $$y^{2}$$
• $$\mathcal{L}=\{l_0,...,l_{5}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto v(0,0)$$

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

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

$$\displaystyle \phi_{1} = \frac{x^{2}}{2} - x y + \frac{x}{2} - \frac{y^{2}}{2} + \frac{y}{2}$$

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

$$\displaystyle \phi_{2} = - \frac{x^{2}}{2} - x y + \frac{x}{2} + \frac{y^{2}}{2} + \frac{y}{2}$$

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

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

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

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

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

$$\displaystyle \phi_{5} = y \left(1 - y\right)$$

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: $$1$$, $$x$$, $$y$$, $$z$$
• $$\mathcal{L}=\{l_0,...,l_{3}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\mathbf{V}\mapsto\displaystyle\int_{f_{0}}(\tfrac{\sqrt{3}}{3})v$$

$$\displaystyle \phi_{0} = 6 x + 6 y + 6 z - 4$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{1}:\mathbf{V}\mapsto\displaystyle\int_{f_{1}}v$$

$$\displaystyle \phi_{1} = 2 - 6 x$$

This DOF is associated with face 1 of the reference element.
$$\displaystyle l_{2}:\mathbf{V}\mapsto\displaystyle\int_{f_{2}}v$$

$$\displaystyle \phi_{2} = 2 - 6 y$$

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{3}:\mathbf{V}\mapsto\displaystyle\int_{f_{3}}v$$

$$\displaystyle \phi_{3} = 2 - 6 z$$

This DOF is associated with face 3 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: $$1$$, $$x$$, $$x^{2}$$, $$y$$, $$x y$$, $$y^{2}$$, $$z$$, $$x z$$, $$y z$$, $$z^{2}$$
• $$\mathcal{L}=\{l_0,...,l_{9}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\mathbf{V}\mapsto\displaystyle\int_{e_{0}}(\tfrac{\sqrt{2}}{2})v$$

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

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{1}:\mathbf{V}\mapsto\displaystyle\int_{e_{1}}(\tfrac{\sqrt{2}}{2})v$$

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

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{2}:\mathbf{V}\mapsto\displaystyle\int_{e_{2}}(\tfrac{\sqrt{2}}{2})v$$

$$\displaystyle \phi_{2} = x^{2} + 2 x y - 4 x z + \frac{2 x}{3} + y^{2} - 4 y z + \frac{2 y}{3} - 2 z^{2} + \frac{8 z}{3} - \frac{2}{3}$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{3}:\mathbf{V}\mapsto\displaystyle\int_{e_{3}}v$$

$$\displaystyle \phi_{3} = 6 x y - 2 x - 2 y + 1$$

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{4}:\mathbf{V}\mapsto\displaystyle\int_{e_{4}}v$$

$$\displaystyle \phi_{4} = 6 x z - 2 x - 2 z + 1$$

This DOF is associated with edge 4 of the reference element.
$$\displaystyle l_{5}:\mathbf{V}\mapsto\displaystyle\int_{e_{5}}v$$

$$\displaystyle \phi_{5} = 6 y z - 2 y - 2 z + 1$$

This DOF is associated with edge 5 of the reference element.
$$\displaystyle l_{6}:\mathbf{V}\mapsto\displaystyle\frac{\sqrt{3}}{3}\int_{f_{0}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle \frac{\sqrt{3}}{3}\\\displaystyle \frac{\sqrt{3}}{3}\\\displaystyle \frac{\sqrt{3}}{3}\end{array}\right)}v$$

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

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{7}:\mathbf{V}\mapsto\displaystyle\int_{f_{1}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)}v$$

$$\displaystyle \phi_{7} = x \left(2 - 3 x\right)$$

This DOF is associated with face 1 of the reference element.
$$\displaystyle l_{8}:\mathbf{V}\mapsto\displaystyle\int_{f_{2}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle 0\\\displaystyle -1\\\displaystyle 0\end{array}\right)}v$$

$$\displaystyle \phi_{8} = y \left(3 y - 2\right)$$

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{9}:\mathbf{V}\mapsto\displaystyle\int_{f_{3}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)}v$$

$$\displaystyle \phi_{9} = z \left(2 - 3 z\right)$$

This DOF is associated with face 3 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: $$1$$, $$x$$, $$x^{2}$$, $$x^{3}$$, $$y$$, $$x y$$, $$x^{2} y$$, $$y^{2}$$, $$x y^{2}$$, $$y^{3}$$, $$z$$, $$x z$$, $$x^{2} z$$, $$y z$$, $$x y z$$, $$y^{2} z$$, $$z^{2}$$, $$x z^{2}$$, $$y z^{2}$$, $$z^{3}$$
• $$\mathcal{L}=\{l_0,...,l_{19}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto v(0,0,0)$$

$$\displaystyle \phi_{0} = - 6 x y z + 2 x y + 2 x z - x + 2 y z - y - z + 1$$

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

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

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

$$\displaystyle \phi_{2} = - \frac{x^{3}}{6} - \frac{x^{2} y}{2} + x^{2} z - \frac{x^{2}}{6} - \frac{x y^{2}}{2} + 2 x y z - \frac{x y}{3} + x z^{2} - \frac{4 x z}{3} + \frac{x}{3} + \frac{y^{3}}{3} - \frac{y^{2} z}{2} + \frac{y^{2}}{3} - \frac{y z^{2}}{2} - \frac{y z}{3} + \frac{y}{3} - \frac{z^{3}}{6} - \frac{z^{2}}{6} + \frac{z}{3}$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{3}:v\mapsto v(0,0,1)$$

$$\displaystyle \phi_{3} = - \frac{x^{3}}{6} + x^{2} y - \frac{x^{2} z}{2} - \frac{x^{2}}{6} + x y^{2} + 2 x y z - \frac{4 x y}{3} - \frac{x z^{2}}{2} - \frac{x z}{3} + \frac{x}{3} - \frac{y^{3}}{6} - \frac{y^{2} z}{2} - \frac{y^{2}}{6} - \frac{y z^{2}}{2} - \frac{y z}{3} + \frac{y}{3} + \frac{z^{3}}{3} + \frac{z^{2}}{3} + \frac{z}{3}$$

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

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

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

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

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{6}:\mathbf{V}\mapsto\displaystyle\frac{\sqrt{2}}{2}\int_{e_{1}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle \frac{\sqrt{3}}{3}\\\displaystyle \frac{\sqrt{3}}{3}\\\displaystyle \frac{\sqrt{3}}{3}\end{array}\right)}v$$

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

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{7}:\mathbf{V}\mapsto\displaystyle\frac{\sqrt{2}}{2}\int_{e_{1}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle 0\\\displaystyle -1\\\displaystyle 0\end{array}\right)}v$$

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

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{8}:\mathbf{V}\mapsto\displaystyle\frac{\sqrt{2}}{2}\int_{e_{2}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle \frac{\sqrt{3}}{3}\\\displaystyle \frac{\sqrt{3}}{3}\\\displaystyle \frac{\sqrt{3}}{3}\end{array}\right)}v$$

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

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{9}:\mathbf{V}\mapsto\displaystyle\frac{\sqrt{2}}{2}\int_{e_{2}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)}v$$

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

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{10}:\mathbf{V}\mapsto\displaystyle\int_{e_{3}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)}v$$

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

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{11}:\mathbf{V}\mapsto\displaystyle\int_{e_{3}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle 0\\\displaystyle -1\\\displaystyle 0\end{array}\right)}v$$

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

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{12}:\mathbf{V}\mapsto\displaystyle\int_{e_{4}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)}v$$

$$\displaystyle \phi_{12} = x \left(3 x z - x - 2 z + 1\right)$$

This DOF is associated with edge 4 of the reference element.
$$\displaystyle l_{13}:\mathbf{V}\mapsto\displaystyle\int_{e_{4}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)}v$$

$$\displaystyle \phi_{13} = z \left(3 x z - 2 x - z + 1\right)$$

This DOF is associated with edge 4 of the reference element.
$$\displaystyle l_{14}:\mathbf{V}\mapsto\displaystyle\int_{e_{5}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle 0\\\displaystyle -1\\\displaystyle 0\end{array}\right)}v$$

$$\displaystyle \phi_{14} = y \left(- 3 y z + y + 2 z - 1\right)$$

This DOF is associated with edge 5 of the reference element.
$$\displaystyle l_{15}:\mathbf{V}\mapsto\displaystyle\int_{e_{5}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)}v$$

$$\displaystyle \phi_{15} = z \left(3 y z - 2 y - z + 1\right)$$

This DOF is associated with edge 5 of the reference element.
$$\displaystyle l_{16}:\mathbf{V}\mapsto\displaystyle\frac{\sqrt{3}}{3}\int_{f_{0}}\frac{\partial^{2}}{\partial\left(\begin{array}{c}\displaystyle \frac{\sqrt{3}}{3}\\\displaystyle \frac{\sqrt{3}}{3}\\\displaystyle \frac{\sqrt{3}}{3}\end{array}\right)^{2}}v$$

$$\displaystyle \phi_{16} = \frac{x^{3}}{3} + x^{2} y + x^{2} z - \frac{2 x^{2}}{3} + x y^{2} + 2 x y z - \frac{4 x y}{3} + x z^{2} - \frac{4 x z}{3} + \frac{x}{3} + \frac{y^{3}}{3} + y^{2} z - \frac{2 y^{2}}{3} + y z^{2} - \frac{4 y z}{3} + \frac{y}{3} + \frac{z^{3}}{3} - \frac{2 z^{2}}{3} + \frac{z}{3}$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{17}:\mathbf{V}\mapsto\displaystyle\int_{f_{1}}\frac{\partial^{2}}{\partial\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)^{2}}v$$

$$\displaystyle \phi_{17} = x^{2} \left(1 - x\right)$$

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

$$\displaystyle \phi_{18} = y^{2} \left(1 - y\right)$$

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

$$\displaystyle \phi_{19} = z^{2} \left(1 - z\right)$$

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

## References

• Wang, M. and Xu, J. Minimal finite element spaces for 2m-th-order partial differential equations in Rn, Mathematics of computation 82(281), 25–43, 2013. [DOI: 10.1090/S0025-5718-2012-02611-1] [BibTeX]