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Degree 3 Morley–Wang–Xu on a tetrahedron
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- \(R\) 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.
\(\displaystyle \phi_{0} = - 6 x y z + 2 x y + 2 x z - x + 2 y z - y - z + 1\)
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.
\(\displaystyle \phi_{1} = \frac{x^{3}}{3} - \frac{x^{2} y}{2} - \frac{x^{2} z}{2} + \frac{x^{2}}{3} - \frac{x y^{2}}{2} + 2 x y z - \frac{x y}{3} - \frac{x z^{2}}{2} - \frac{x z}{3} + \frac{x}{3} - \frac{y^{3}}{6} + y^{2} z - \frac{y^{2}}{6} + y z^{2} - \frac{4 y z}{3} + \frac{y}{3} - \frac{z^{3}}{6} - \frac{z^{2}}{6} + \frac{z}{3}\)
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.
\(\displaystyle \phi_{2} = - \frac{x^{3}}{6} - \frac{x^{2} y}{2} + x^{2} z - \frac{x^{2}}{6} - \frac{x y^{2}}{2} + 2 x y z - \frac{x y}{3} + x z^{2} - \frac{4 x z}{3} + \frac{x}{3} + \frac{y^{3}}{3} - \frac{y^{2} z}{2} + \frac{y^{2}}{3} - \frac{y z^{2}}{2} - \frac{y z}{3} + \frac{y}{3} - \frac{z^{3}}{6} - \frac{z^{2}}{6} + \frac{z}{3}\)
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.
\(\displaystyle \phi_{3} = - \frac{x^{3}}{6} + x^{2} y - \frac{x^{2} z}{2} - \frac{x^{2}}{6} + x y^{2} + 2 x y z - \frac{4 x y}{3} - \frac{x z^{2}}{2} - \frac{x z}{3} + \frac{x}{3} - \frac{y^{3}}{6} - \frac{y^{2} z}{2} - \frac{y^{2}}{6} - \frac{y z^{2}}{2} - \frac{y z}{3} + \frac{y}{3} + \frac{z^{3}}{3} + \frac{z^{2}}{3} + \frac{z}{3}\)
This DOF is associated with vertex 3 of the reference element.
\(\displaystyle l_{4}:\mathbf{V}\mapsto\displaystyle \frac{\sqrt{2}}{2}\int_{e_{0}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle \frac{\sqrt{3}}{3}\\\displaystyle \frac{\sqrt{3}}{3}\\\displaystyle \frac{\sqrt{3}}{3}\end{array}\right)}v\)
where \(e_{0}\) is the 0th edge.
\(\displaystyle \phi_{4} = \frac{\sqrt{3} \left(- 4 x^{3} - 12 x^{2} y - 12 x^{2} z + 8 x^{2} - 12 x y^{2} - 24 x y z + 16 x y - 12 x z^{2} + 16 x z - 4 x + 5 y^{3} + 15 y^{2} z - y^{2} + 15 y z^{2} - 2 y z - 4 y + 5 z^{3} - z^{2} - 4 z\right)}{18}\)
This DOF is associated with edge 0 of the reference element.
where \(e_{0}\) is the 0th edge.
\(\displaystyle \phi_{4} = \frac{\sqrt{3} \left(- 4 x^{3} - 12 x^{2} y - 12 x^{2} z + 8 x^{2} - 12 x y^{2} - 24 x y z + 16 x y - 12 x z^{2} + 16 x z - 4 x + 5 y^{3} + 15 y^{2} z - y^{2} + 15 y z^{2} - 2 y z - 4 y + 5 z^{3} - z^{2} - 4 z\right)}{18}\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{5}:\mathbf{V}\mapsto\displaystyle \frac{\sqrt{2}}{2}\int_{e_{0}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)}v\)
where \(e_{0}\) is the 0th edge.
\(\displaystyle \phi_{5} = - \frac{2 x^{3}}{3} - 2 x^{2} y - 2 x^{2} z + \frac{4 x^{2}}{3} + x y^{2} + 2 x y z + \frac{2 x y}{3} + x z^{2} + \frac{2 x z}{3} - \frac{2 x}{3} - \frac{y^{3}}{6} - \frac{y^{2} z}{2} - \frac{y^{2}}{6} - \frac{y z^{2}}{2} - \frac{y z}{3} + \frac{y}{3} - \frac{z^{3}}{6} - \frac{z^{2}}{6} + \frac{z}{3}\)
This DOF is associated with edge 0 of the reference element.
where \(e_{0}\) is the 0th edge.
\(\displaystyle \phi_{5} = - \frac{2 x^{3}}{3} - 2 x^{2} y - 2 x^{2} z + \frac{4 x^{2}}{3} + x y^{2} + 2 x y z + \frac{2 x y}{3} + x z^{2} + \frac{2 x z}{3} - \frac{2 x}{3} - \frac{y^{3}}{6} - \frac{y^{2} z}{2} - \frac{y^{2}}{6} - \frac{y z^{2}}{2} - \frac{y z}{3} + \frac{y}{3} - \frac{z^{3}}{6} - \frac{z^{2}}{6} + \frac{z}{3}\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{6}:\mathbf{V}\mapsto\displaystyle \frac{\sqrt{2}}{2}\int_{e_{1}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle \frac{\sqrt{3}}{3}\\\displaystyle \frac{\sqrt{3}}{3}\\\displaystyle \frac{\sqrt{3}}{3}\end{array}\right)}v\)
where \(e_{1}\) is the 1st edge.
\(\displaystyle \phi_{6} = \frac{\sqrt{3} \left(5 x^{3} - 12 x^{2} y + 15 x^{2} z - x^{2} - 12 x y^{2} - 24 x y z + 16 x y + 15 x z^{2} - 2 x z - 4 x - 4 y^{3} - 12 y^{2} z + 8 y^{2} - 12 y z^{2} + 16 y z - 4 y + 5 z^{3} - z^{2} - 4 z\right)}{18}\)
This DOF is associated with edge 1 of the reference element.
where \(e_{1}\) is the 1st edge.
\(\displaystyle \phi_{6} = \frac{\sqrt{3} \left(5 x^{3} - 12 x^{2} y + 15 x^{2} z - x^{2} - 12 x y^{2} - 24 x y z + 16 x y + 15 x z^{2} - 2 x z - 4 x - 4 y^{3} - 12 y^{2} z + 8 y^{2} - 12 y z^{2} + 16 y z - 4 y + 5 z^{3} - z^{2} - 4 z\right)}{18}\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{7}:\mathbf{V}\mapsto\displaystyle \frac{\sqrt{2}}{2}\int_{e_{1}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle 0\\\displaystyle -1\\\displaystyle 0\end{array}\right)}v\)
where \(e_{1}\) is the 1st edge.
\(\displaystyle \phi_{7} = \frac{x^{3}}{6} - x^{2} y + \frac{x^{2} z}{2} + \frac{x^{2}}{6} + 2 x y^{2} - 2 x y z - \frac{2 x y}{3} + \frac{x z^{2}}{2} + \frac{x z}{3} - \frac{x}{3} + \frac{2 y^{3}}{3} + 2 y^{2} z - \frac{4 y^{2}}{3} - y z^{2} - \frac{2 y z}{3} + \frac{2 y}{3} + \frac{z^{3}}{6} + \frac{z^{2}}{6} - \frac{z}{3}\)
This DOF is associated with edge 1 of the reference element.
where \(e_{1}\) is the 1st edge.
\(\displaystyle \phi_{7} = \frac{x^{3}}{6} - x^{2} y + \frac{x^{2} z}{2} + \frac{x^{2}}{6} + 2 x y^{2} - 2 x y z - \frac{2 x y}{3} + \frac{x z^{2}}{2} + \frac{x z}{3} - \frac{x}{3} + \frac{2 y^{3}}{3} + 2 y^{2} z - \frac{4 y^{2}}{3} - y z^{2} - \frac{2 y z}{3} + \frac{2 y}{3} + \frac{z^{3}}{6} + \frac{z^{2}}{6} - \frac{z}{3}\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{8}:\mathbf{V}\mapsto\displaystyle \frac{\sqrt{2}}{2}\int_{e_{2}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle \frac{\sqrt{3}}{3}\\\displaystyle \frac{\sqrt{3}}{3}\\\displaystyle \frac{\sqrt{3}}{3}\end{array}\right)}v\)
where \(e_{2}\) is the 2nd edge.
\(\displaystyle \phi_{8} = \frac{\sqrt{3} \left(5 x^{3} + 15 x^{2} y - 12 x^{2} z - x^{2} + 15 x y^{2} - 24 x y z - 2 x y - 12 x z^{2} + 16 x z - 4 x + 5 y^{3} - 12 y^{2} z - y^{2} - 12 y z^{2} + 16 y z - 4 y - 4 z^{3} + 8 z^{2} - 4 z\right)}{18}\)
This DOF is associated with edge 2 of the reference element.
where \(e_{2}\) is the 2nd edge.
\(\displaystyle \phi_{8} = \frac{\sqrt{3} \left(5 x^{3} + 15 x^{2} y - 12 x^{2} z - x^{2} + 15 x y^{2} - 24 x y z - 2 x y - 12 x z^{2} + 16 x z - 4 x + 5 y^{3} - 12 y^{2} z - y^{2} - 12 y z^{2} + 16 y z - 4 y - 4 z^{3} + 8 z^{2} - 4 z\right)}{18}\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{9}:\mathbf{V}\mapsto\displaystyle \frac{\sqrt{2}}{2}\int_{e_{2}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)}v\)
where \(e_{2}\) is the 2nd edge.
\(\displaystyle \phi_{9} = - \frac{x^{3}}{6} - \frac{x^{2} y}{2} + x^{2} z - \frac{x^{2}}{6} - \frac{x y^{2}}{2} + 2 x y z - \frac{x y}{3} - 2 x z^{2} + \frac{2 x z}{3} + \frac{x}{3} - \frac{y^{3}}{6} + y^{2} z - \frac{y^{2}}{6} - 2 y z^{2} + \frac{2 y z}{3} + \frac{y}{3} - \frac{2 z^{3}}{3} + \frac{4 z^{2}}{3} - \frac{2 z}{3}\)
This DOF is associated with edge 2 of the reference element.
where \(e_{2}\) is the 2nd edge.
\(\displaystyle \phi_{9} = - \frac{x^{3}}{6} - \frac{x^{2} y}{2} + x^{2} z - \frac{x^{2}}{6} - \frac{x y^{2}}{2} + 2 x y z - \frac{x y}{3} - 2 x z^{2} + \frac{2 x z}{3} + \frac{x}{3} - \frac{y^{3}}{6} + y^{2} z - \frac{y^{2}}{6} - 2 y z^{2} + \frac{2 y z}{3} + \frac{y}{3} - \frac{2 z^{3}}{3} + \frac{4 z^{2}}{3} - \frac{2 z}{3}\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{10}:\mathbf{V}\mapsto\displaystyle \int_{e_{3}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)}v\)
where \(e_{3}\) is the 3rd edge.
\(\displaystyle \phi_{10} = x \left(3 x y - x - 2 y + 1\right)\)
This DOF is associated with edge 3 of the reference element.
where \(e_{3}\) is the 3rd edge.
\(\displaystyle \phi_{10} = x \left(3 x y - x - 2 y + 1\right)\)
This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{11}:\mathbf{V}\mapsto\displaystyle \int_{e_{3}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle 0\\\displaystyle -1\\\displaystyle 0\end{array}\right)}v\)
where \(e_{3}\) is the 3rd edge.
\(\displaystyle \phi_{11} = y \left(- 3 x y + 2 x + y - 1\right)\)
This DOF is associated with edge 3 of the reference element.
where \(e_{3}\) is the 3rd edge.
\(\displaystyle \phi_{11} = y \left(- 3 x y + 2 x + y - 1\right)\)
This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{12}:\mathbf{V}\mapsto\displaystyle \int_{e_{4}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)}v\)
where \(e_{4}\) is the 4th edge.
\(\displaystyle \phi_{12} = x \left(3 x z - x - 2 z + 1\right)\)
This DOF is associated with edge 4 of the reference element.
where \(e_{4}\) is the 4th edge.
\(\displaystyle \phi_{12} = x \left(3 x z - x - 2 z + 1\right)\)
This DOF is associated with edge 4 of the reference element.
\(\displaystyle l_{13}:\mathbf{V}\mapsto\displaystyle \int_{e_{4}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)}v\)
where \(e_{4}\) is the 4th edge.
\(\displaystyle \phi_{13} = z \left(3 x z - 2 x - z + 1\right)\)
This DOF is associated with edge 4 of the reference element.
where \(e_{4}\) is the 4th edge.
\(\displaystyle \phi_{13} = z \left(3 x z - 2 x - z + 1\right)\)
This DOF is associated with edge 4 of the reference element.
\(\displaystyle l_{14}:\mathbf{V}\mapsto\displaystyle \int_{e_{5}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle 0\\\displaystyle -1\\\displaystyle 0\end{array}\right)}v\)
where \(e_{5}\) is the 5th edge.
\(\displaystyle \phi_{14} = y \left(- 3 y z + y + 2 z - 1\right)\)
This DOF is associated with edge 5 of the reference element.
where \(e_{5}\) is the 5th edge.
\(\displaystyle \phi_{14} = y \left(- 3 y z + y + 2 z - 1\right)\)
This DOF is associated with edge 5 of the reference element.
\(\displaystyle l_{15}:\mathbf{V}\mapsto\displaystyle \int_{e_{5}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)}v\)
where \(e_{5}\) is the 5th edge.
\(\displaystyle \phi_{15} = z \left(3 y z - 2 y - z + 1\right)\)
This DOF is associated with edge 5 of the reference element.
where \(e_{5}\) is the 5th edge.
\(\displaystyle \phi_{15} = z \left(3 y z - 2 y - z + 1\right)\)
This DOF is associated with edge 5 of the reference element.
\(\displaystyle l_{16}:\mathbf{V}\mapsto\displaystyle \frac{\sqrt{3}}{3}\int_{f_{0}}\frac{\partial^{2}}{\partial\left(\begin{array}{c}\displaystyle \frac{\sqrt{3}}{3}\\\displaystyle \frac{\sqrt{3}}{3}\\\displaystyle \frac{\sqrt{3}}{3}\end{array}\right)^{2}}v\)
where \(f_{0}\) is the 0th face.
\(\displaystyle \phi_{16} = \frac{x^{3}}{3} + x^{2} y + x^{2} z - \frac{2 x^{2}}{3} + x y^{2} + 2 x y z - \frac{4 x y}{3} + x z^{2} - \frac{4 x z}{3} + \frac{x}{3} + \frac{y^{3}}{3} + y^{2} z - \frac{2 y^{2}}{3} + y z^{2} - \frac{4 y z}{3} + \frac{y}{3} + \frac{z^{3}}{3} - \frac{2 z^{2}}{3} + \frac{z}{3}\)
This DOF is associated with face 0 of the reference element.
where \(f_{0}\) is the 0th face.
\(\displaystyle \phi_{16} = \frac{x^{3}}{3} + x^{2} y + x^{2} z - \frac{2 x^{2}}{3} + x y^{2} + 2 x y z - \frac{4 x y}{3} + x z^{2} - \frac{4 x z}{3} + \frac{x}{3} + \frac{y^{3}}{3} + y^{2} z - \frac{2 y^{2}}{3} + y z^{2} - \frac{4 y z}{3} + \frac{y}{3} + \frac{z^{3}}{3} - \frac{2 z^{2}}{3} + \frac{z}{3}\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{17}:\mathbf{V}\mapsto\displaystyle \int_{f_{1}}\frac{\partial^{2}}{\partial\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)^{2}}v\)
where \(f_{1}\) is the 1st face.
\(\displaystyle \phi_{17} = x^{2} \left(1 - x\right)\)
This DOF is associated with face 1 of the reference element.
where \(f_{1}\) is the 1st face.
\(\displaystyle \phi_{17} = x^{2} \left(1 - x\right)\)
This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{18}:\mathbf{V}\mapsto\displaystyle \int_{f_{2}}\frac{\partial^{2}}{\partial\left(\begin{array}{c}\displaystyle 0\\\displaystyle -1\\\displaystyle 0\end{array}\right)^{2}}v\)
where \(f_{2}\) is the 2nd face.
\(\displaystyle \phi_{18} = y^{2} \left(1 - y\right)\)
This DOF is associated with face 2 of the reference element.
where \(f_{2}\) is the 2nd face.
\(\displaystyle \phi_{18} = y^{2} \left(1 - y\right)\)
This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{19}:\mathbf{V}\mapsto\displaystyle \int_{f_{3}}\frac{\partial^{2}}{\partial\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)^{2}}v\)
where \(f_{3}\) is the 3rd face.
\(\displaystyle \phi_{19} = z^{2} \left(1 - z\right)\)
This DOF is associated with face 3 of the reference element.
where \(f_{3}\) is the 3rd face.
\(\displaystyle \phi_{19} = z^{2} \left(1 - z\right)\)
This DOF is associated with face 3 of the reference element.