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Morley–Wang–Xu

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Orders\(1\leqslant k\leqslant {'interval': 1, 'triangle': 2, 'tetrahedron': 3}\)
Reference elementsinterval, triangle, tetrahedron
Polynomial set\(\mathcal{P}_{k}\)
↓ Show polynomial set definitions ↓
DOFsOn 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 DOFsinterval: \(k+1\) (A000027)
triangle: \((k+1)(k+2)/2\) (A000217)
tetrahedron: \((k+1)(k+2)(k+3)/6\) (A000292)
CategoriesScalar-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