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Degree 3 conforming Crouzeix–Raviart 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: \(1\), \(x\), \(x^{2}\), \(x^{3}\), \(y\), \(x y\), \(x^{2} y\), \(y^{2}\), \(x y^{2}\), \(y^{3}\), \(x y^{2} \left(x + y\right)\), \(x^{2} y \left(x + y\right)\)
- \(\mathcal{L}=\{l_0,...,l_{11}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0,0)\)
\(\displaystyle \phi_{0} = \frac{243 x^{3} y}{20} - \frac{9 x^{3}}{2} + \frac{243 x^{2} y^{2}}{10} - \frac{333 x^{2} y}{10} + 9 x^{2} + \frac{243 x y^{3}}{20} - \frac{333 x y^{2}}{10} + \frac{513 x y}{20} - \frac{11 x}{2} - \frac{9 y^{3}}{2} + 9 y^{2} - \frac{11 y}{2} + 1\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \phi_{0} = \frac{243 x^{3} y}{20} - \frac{9 x^{3}}{2} + \frac{243 x^{2} y^{2}}{10} - \frac{333 x^{2} y}{10} + 9 x^{2} + \frac{243 x y^{3}}{20} - \frac{333 x y^{2}}{10} + \frac{513 x y}{20} - \frac{11 x}{2} - \frac{9 y^{3}}{2} + 9 y^{2} - \frac{11 y}{2} + 1\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:v\mapsto v(1,0)\)
\(\displaystyle \phi_{1} = \frac{x \left(- 243 x^{2} y + 90 x^{2} - 243 x y^{2} + 333 x y - 90 x + 90 y^{2} - 90 y + 20\right)}{20}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \phi_{1} = \frac{x \left(- 243 x^{2} y + 90 x^{2} - 243 x y^{2} + 333 x y - 90 x + 90 y^{2} - 90 y + 20\right)}{20}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{2}:v\mapsto v(0,1)\)
\(\displaystyle \phi_{2} = \frac{y \left(- 243 x^{2} y + 90 x^{2} - 243 x y^{2} + 333 x y - 90 x + 90 y^{2} - 90 y + 20\right)}{20}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \phi_{2} = \frac{y \left(- 243 x^{2} y + 90 x^{2} - 243 x y^{2} + 333 x y - 90 x + 90 y^{2} - 90 y + 20\right)}{20}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{3}:v\mapsto v(\tfrac{2}{3},\tfrac{1}{3})\)
\(\displaystyle \phi_{3} = \frac{9 x y \left(108 x^{2} + 81 x y - 102 x - 27 y^{2} + 3 y + 14\right)}{20}\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle \phi_{3} = \frac{9 x y \left(108 x^{2} + 81 x y - 102 x - 27 y^{2} + 3 y + 14\right)}{20}\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{4}:v\mapsto v(\tfrac{1}{3},\tfrac{2}{3})\)
\(\displaystyle \phi_{4} = \frac{9 x y \left(- 27 x^{2} + 81 x y + 3 x + 108 y^{2} - 102 y + 14\right)}{20}\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle \phi_{4} = \frac{9 x y \left(- 27 x^{2} + 81 x y + 3 x + 108 y^{2} - 102 y + 14\right)}{20}\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{5}:v\mapsto v(0,\tfrac{1}{3})\)
\(\displaystyle \phi_{5} = \frac{9 y \left(- 108 x^{3} - 243 x^{2} y + 222 x^{2} - 135 x y^{2} + 279 x y - 134 x + 30 y^{2} - 50 y + 20\right)}{20}\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle \phi_{5} = \frac{9 y \left(- 108 x^{3} - 243 x^{2} y + 222 x^{2} - 135 x y^{2} + 279 x y - 134 x + 30 y^{2} - 50 y + 20\right)}{20}\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{6}:v\mapsto v(0,\tfrac{2}{3})\)
\(\displaystyle \phi_{6} = \frac{9 y \left(27 x^{3} + 162 x^{2} y - 78 x^{2} + 135 x y^{2} - 216 x y + 61 x - 30 y^{2} + 40 y - 10\right)}{20}\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle \phi_{6} = \frac{9 y \left(27 x^{3} + 162 x^{2} y - 78 x^{2} + 135 x y^{2} - 216 x y + 61 x - 30 y^{2} + 40 y - 10\right)}{20}\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{7}:v\mapsto v(\tfrac{1}{3},0)\)
\(\displaystyle \phi_{7} = \frac{9 x \left(- 135 x^{2} y + 30 x^{2} - 243 x y^{2} + 279 x y - 50 x - 108 y^{3} + 222 y^{2} - 134 y + 20\right)}{20}\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle \phi_{7} = \frac{9 x \left(- 135 x^{2} y + 30 x^{2} - 243 x y^{2} + 279 x y - 50 x - 108 y^{3} + 222 y^{2} - 134 y + 20\right)}{20}\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{8}:v\mapsto v(\tfrac{2}{3},0)\)
\(\displaystyle \phi_{8} = \frac{9 x \left(135 x^{2} y - 30 x^{2} + 162 x y^{2} - 216 x y + 40 x + 27 y^{3} - 78 y^{2} + 61 y - 10\right)}{20}\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle \phi_{8} = \frac{9 x \left(135 x^{2} y - 30 x^{2} + 162 x y^{2} - 216 x y + 40 x + 27 y^{3} - 78 y^{2} + 61 y - 10\right)}{20}\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{9}:v\mapsto v(\tfrac{2}{9},\tfrac{2}{9})\)
\(\displaystyle \phi_{9} = \frac{243 x y \left(9 x^{2} + 18 x y - 16 x + 9 y^{2} - 16 y + 7\right)}{20}\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle \phi_{9} = \frac{243 x y \left(9 x^{2} + 18 x y - 16 x + 9 y^{2} - 16 y + 7\right)}{20}\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{10}:v\mapsto v(\tfrac{2}{9},\tfrac{5}{9})\)
\(\displaystyle \phi_{10} = \frac{243 x y \left(- 9 x y + 2 x - 9 y^{2} + 11 y - 2\right)}{20}\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle \phi_{10} = \frac{243 x y \left(- 9 x y + 2 x - 9 y^{2} + 11 y - 2\right)}{20}\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{11}:v\mapsto v(\tfrac{5}{9},\tfrac{2}{9})\)
\(\displaystyle \phi_{11} = \frac{243 x y \left(- 9 x^{2} - 9 x y + 11 x + 2 y - 2\right)}{20}\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle \phi_{11} = \frac{243 x y \left(- 9 x^{2} - 9 x y + 11 x + 2 y - 2\right)}{20}\)
This DOF is associated with face 0 of the reference element.