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Degree 3 Lagrange 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: \(1\), \(x\), \(x^{2}\), \(x^{3}\), \(y\), \(x y\), \(x^{2} y\), \(y^{2}\), \(x y^{2}\), \(y^{3}\)
- \(\mathcal{L}=\{l_0,...,l_{9}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0,0)\)
\(\displaystyle \phi_{0} = - \frac{9 x^{3}}{2} - \frac{27 x^{2} y}{2} + 9 x^{2} - \frac{27 x y^{2}}{2} + 18 x y - \frac{11 x}{2} - \frac{9 y^{3}}{2} + 9 y^{2} - \frac{11 y}{2} + 1\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \phi_{0} = - \frac{9 x^{3}}{2} - \frac{27 x^{2} y}{2} + 9 x^{2} - \frac{27 x y^{2}}{2} + 18 x y - \frac{11 x}{2} - \frac{9 y^{3}}{2} + 9 y^{2} - \frac{11 y}{2} + 1\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:v\mapsto v(1,0)\)
\(\displaystyle \phi_{1} = \frac{x \left(9 x^{2} - 9 x + 2\right)}{2}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \phi_{1} = \frac{x \left(9 x^{2} - 9 x + 2\right)}{2}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{2}:v\mapsto v(0,1)\)
\(\displaystyle \phi_{2} = \frac{y \left(9 y^{2} - 9 y + 2\right)}{2}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \phi_{2} = \frac{y \left(9 y^{2} - 9 y + 2\right)}{2}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{3}:v\mapsto v(\tfrac{2}{3},\tfrac{1}{3})\)
\(\displaystyle \phi_{3} = \frac{9 x y \left(3 x - 1\right)}{2}\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle \phi_{3} = \frac{9 x y \left(3 x - 1\right)}{2}\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{4}:v\mapsto v(\tfrac{1}{3},\tfrac{2}{3})\)
\(\displaystyle \phi_{4} = \frac{9 x y \left(3 y - 1\right)}{2}\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle \phi_{4} = \frac{9 x y \left(3 y - 1\right)}{2}\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{5}:v\mapsto v(0,\tfrac{1}{3})\)
\(\displaystyle \phi_{5} = \frac{9 y \left(3 x^{2} + 6 x y - 5 x + 3 y^{2} - 5 y + 2\right)}{2}\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle \phi_{5} = \frac{9 y \left(3 x^{2} + 6 x y - 5 x + 3 y^{2} - 5 y + 2\right)}{2}\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{6}:v\mapsto v(0,\tfrac{2}{3})\)
\(\displaystyle \phi_{6} = \frac{9 y \left(- 3 x y + x - 3 y^{2} + 4 y - 1\right)}{2}\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle \phi_{6} = \frac{9 y \left(- 3 x y + x - 3 y^{2} + 4 y - 1\right)}{2}\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{7}:v\mapsto v(\tfrac{1}{3},0)\)
\(\displaystyle \phi_{7} = \frac{9 x \left(3 x^{2} + 6 x y - 5 x + 3 y^{2} - 5 y + 2\right)}{2}\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle \phi_{7} = \frac{9 x \left(3 x^{2} + 6 x y - 5 x + 3 y^{2} - 5 y + 2\right)}{2}\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{8}:v\mapsto v(\tfrac{2}{3},0)\)
\(\displaystyle \phi_{8} = \frac{9 x \left(- 3 x^{2} - 3 x y + 4 x + y - 1\right)}{2}\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle \phi_{8} = \frac{9 x \left(- 3 x^{2} - 3 x y + 4 x + y - 1\right)}{2}\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{9}:v\mapsto v(\tfrac{1}{3},\tfrac{1}{3})\)
\(\displaystyle \phi_{9} = 27 x y \left(- x - y + 1\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle \phi_{9} = 27 x y \left(- x - y + 1\right)\)
This DOF is associated with face 0 of the reference element.