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Degree 3 Wu–Xu on a triangle
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- \(R\) is the reference triangle. The following numbering of the subentities of the reference is used:
- \(\mathcal{V}\) is spanned by: \(1\), \(x\), \(x^{2}\), \(x^{3}\), \(y\), \(x y\), \(x^{2} y\), \(y^{2}\), \(x y^{2}\), \(y^{3}\), \(x^{2} y \left(- x - y + 1\right)\), \(x y^{2} \left(- x - y + 1\right)\)
- \(\mathcal{L}=\{l_0,...,l_{11}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0,0)\)
\(\displaystyle \phi_{0} = 12 x^{3} y + 2 x^{3} + 24 x^{2} y^{2} - 18 x^{2} y - 3 x^{2} + 12 x y^{3} - 18 x y^{2} + 6 x y + 2 y^{3} - 3 y^{2} + 1\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \phi_{0} = 12 x^{3} y + 2 x^{3} + 24 x^{2} y^{2} - 18 x^{2} y - 3 x^{2} + 12 x y^{3} - 18 x y^{2} + 6 x y + 2 y^{3} - 3 y^{2} + 1\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:v\mapsto\frac{\partial}{\partial x}v(0,0)\)
\(\displaystyle \phi_{1} = x \left(- 4 x^{2} y + x^{2} + 6 x y - 2 x + 4 y^{3} - 3 y^{2} - 2 y + 1\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \phi_{1} = x \left(- 4 x^{2} y + x^{2} + 6 x y - 2 x + 4 y^{3} - 3 y^{2} - 2 y + 1\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{2}:v\mapsto\frac{\partial}{\partial y}v(0,0)\)
\(\displaystyle \phi_{2} = y \left(4 x^{3} - 3 x^{2} - 4 x y^{2} + 6 x y - 2 x + y^{2} - 2 y + 1\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \phi_{2} = y \left(4 x^{3} - 3 x^{2} - 4 x y^{2} + 6 x y - 2 x + y^{2} - 2 y + 1\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{3}:v\mapsto v(1,0)\)
\(\displaystyle \phi_{3} = x \left(- 6 x^{2} y - 2 x^{2} - 12 x y^{2} + 9 x y + 3 x - 6 y^{3} + 9 y^{2} - 3 y\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \phi_{3} = x \left(- 6 x^{2} y - 2 x^{2} - 12 x y^{2} + 9 x y + 3 x - 6 y^{3} + 9 y^{2} - 3 y\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{4}:v\mapsto\frac{\partial}{\partial x}v(1,0)\)
\(\displaystyle \phi_{4} = x^{2} \left(x - 1\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \phi_{4} = x^{2} \left(x - 1\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{5}:v\mapsto\frac{\partial}{\partial y}v(1,0)\)
\(\displaystyle \phi_{5} = x y \left(- 4 x^{2} - 12 x y + 9 x - 8 y^{2} + 12 y - 4\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \phi_{5} = x y \left(- 4 x^{2} - 12 x y + 9 x - 8 y^{2} + 12 y - 4\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{6}:v\mapsto v(0,1)\)
\(\displaystyle \phi_{6} = y \left(- 6 x^{3} - 12 x^{2} y + 9 x^{2} - 6 x y^{2} + 9 x y - 3 x - 2 y^{2} + 3 y\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \phi_{6} = y \left(- 6 x^{3} - 12 x^{2} y + 9 x^{2} - 6 x y^{2} + 9 x y - 3 x - 2 y^{2} + 3 y\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{7}:v\mapsto\frac{\partial}{\partial x}v(0,1)\)
\(\displaystyle \phi_{7} = x y \left(- 8 x^{2} - 12 x y + 12 x - 4 y^{2} + 9 y - 4\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \phi_{7} = x y \left(- 8 x^{2} - 12 x y + 12 x - 4 y^{2} + 9 y - 4\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{8}:v\mapsto\frac{\partial}{\partial y}v(0,1)\)
\(\displaystyle \phi_{8} = y^{2} \left(y - 1\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \phi_{8} = y^{2} \left(y - 1\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{9}:\mathbf{V}\mapsto\displaystyle \frac{\sqrt{2}}{2}\int_{e_{0}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle - \frac{\sqrt{2}}{2}\end{array}\right)}v\)
where \(e_{0}\) is the 0th edge.
\(\displaystyle \phi_{9} = 3 \sqrt{2} x y \left(- 2 x^{2} - 4 x y + 3 x - 2 y^{2} + 3 y - 1\right)\)
This DOF is associated with edge 0 of the reference element.
where \(e_{0}\) is the 0th edge.
\(\displaystyle \phi_{9} = 3 \sqrt{2} x y \left(- 2 x^{2} - 4 x y + 3 x - 2 y^{2} + 3 y - 1\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{10}:\mathbf{V}\mapsto\displaystyle \int_{e_{1}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle -1\\\displaystyle 0\end{array}\right)}v\)
where \(e_{1}\) is the 1st edge.
\(\displaystyle \phi_{10} = 6 x y \left(- 2 x^{2} - 2 x y + 3 x + y - 1\right)\)
This DOF is associated with edge 1 of the reference element.
where \(e_{1}\) is the 1st edge.
\(\displaystyle \phi_{10} = 6 x y \left(- 2 x^{2} - 2 x y + 3 x + y - 1\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{11}:\mathbf{V}\mapsto\displaystyle \int_{e_{2}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)}v\)
where \(e_{2}\) is the 2nd edge.
\(\displaystyle \phi_{11} = 6 x y \left(2 x y - x + 2 y^{2} - 3 y + 1\right)\)
This DOF is associated with edge 2 of the reference element.
where \(e_{2}\) is the 2nd edge.
\(\displaystyle \phi_{11} = 6 x y \left(2 x y - x + 2 y^{2} - 3 y + 1\right)\)
This DOF is associated with edge 2 of the reference element.