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Degree 3 Lagrange on a quadrilateral
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- \(R\) is the reference quadrilateral. The following numbering of the subentities of the reference is used:
- \(\mathcal{V}\) is spanned by: \(1\), \(y\), \(y^{2}\), \(y^{3}\), \(x\), \(x y\), \(x y^{2}\), \(x y^{3}\), \(x^{2}\), \(x^{2} y\), \(x^{2} y^{2}\), \(x^{2} y^{3}\), \(x^{3}\), \(x^{3} y\), \(x^{3} y^{2}\), \(x^{3} y^{3}\)
- \(\mathcal{L}=\{l_0,...,l_{15}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0,0)\)
\(\displaystyle \phi_{0} = \frac{81 x^{3} y^{3}}{4} - \frac{81 x^{3} y^{2}}{2} + \frac{99 x^{3} y}{4} - \frac{9 x^{3}}{2} - \frac{81 x^{2} y^{3}}{2} + 81 x^{2} y^{2} - \frac{99 x^{2} y}{2} + 9 x^{2} + \frac{99 x y^{3}}{4} - \frac{99 x y^{2}}{2} + \frac{121 x y}{4} - \frac{11 x}{2} - \frac{9 y^{3}}{2} + 9 y^{2} - \frac{11 y}{2} + 1\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \phi_{0} = \frac{81 x^{3} y^{3}}{4} - \frac{81 x^{3} y^{2}}{2} + \frac{99 x^{3} y}{4} - \frac{9 x^{3}}{2} - \frac{81 x^{2} y^{3}}{2} + 81 x^{2} y^{2} - \frac{99 x^{2} y}{2} + 9 x^{2} + \frac{99 x y^{3}}{4} - \frac{99 x y^{2}}{2} + \frac{121 x y}{4} - \frac{11 x}{2} - \frac{9 y^{3}}{2} + 9 y^{2} - \frac{11 y}{2} + 1\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:v\mapsto v(1,0)\)
\(\displaystyle \phi_{1} = \frac{x \left(- 81 x^{2} y^{3} + 162 x^{2} y^{2} - 99 x^{2} y + 18 x^{2} + 81 x y^{3} - 162 x y^{2} + 99 x y - 18 x - 18 y^{3} + 36 y^{2} - 22 y + 4\right)}{4}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \phi_{1} = \frac{x \left(- 81 x^{2} y^{3} + 162 x^{2} y^{2} - 99 x^{2} y + 18 x^{2} + 81 x y^{3} - 162 x y^{2} + 99 x y - 18 x - 18 y^{3} + 36 y^{2} - 22 y + 4\right)}{4}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{2}:v\mapsto v(0,1)\)
\(\displaystyle \phi_{2} = \frac{y \left(- 81 x^{3} y^{2} + 81 x^{3} y - 18 x^{3} + 162 x^{2} y^{2} - 162 x^{2} y + 36 x^{2} - 99 x y^{2} + 99 x y - 22 x + 18 y^{2} - 18 y + 4\right)}{4}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \phi_{2} = \frac{y \left(- 81 x^{3} y^{2} + 81 x^{3} y - 18 x^{3} + 162 x^{2} y^{2} - 162 x^{2} y + 36 x^{2} - 99 x y^{2} + 99 x y - 22 x + 18 y^{2} - 18 y + 4\right)}{4}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{3}:v\mapsto v(1,1)\)
\(\displaystyle \phi_{3} = \frac{x y \left(81 x^{2} y^{2} - 81 x^{2} y + 18 x^{2} - 81 x y^{2} + 81 x y - 18 x + 18 y^{2} - 18 y + 4\right)}{4}\)
This DOF is associated with vertex 3 of the reference element.
\(\displaystyle \phi_{3} = \frac{x y \left(81 x^{2} y^{2} - 81 x^{2} y + 18 x^{2} - 81 x y^{2} + 81 x y - 18 x + 18 y^{2} - 18 y + 4\right)}{4}\)
This DOF is associated with vertex 3 of the reference element.
\(\displaystyle l_{4}:v\mapsto v(\tfrac{1}{3},0)\)
\(\displaystyle \phi_{4} = \frac{9 x \left(- 27 x^{2} y^{3} + 54 x^{2} y^{2} - 33 x^{2} y + 6 x^{2} + 45 x y^{3} - 90 x y^{2} + 55 x y - 10 x - 18 y^{3} + 36 y^{2} - 22 y + 4\right)}{4}\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle \phi_{4} = \frac{9 x \left(- 27 x^{2} y^{3} + 54 x^{2} y^{2} - 33 x^{2} y + 6 x^{2} + 45 x y^{3} - 90 x y^{2} + 55 x y - 10 x - 18 y^{3} + 36 y^{2} - 22 y + 4\right)}{4}\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{5}:v\mapsto v(\tfrac{2}{3},0)\)
\(\displaystyle \phi_{5} = \frac{9 x \left(27 x^{2} y^{3} - 54 x^{2} y^{2} + 33 x^{2} y - 6 x^{2} - 36 x y^{3} + 72 x y^{2} - 44 x y + 8 x + 9 y^{3} - 18 y^{2} + 11 y - 2\right)}{4}\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle \phi_{5} = \frac{9 x \left(27 x^{2} y^{3} - 54 x^{2} y^{2} + 33 x^{2} y - 6 x^{2} - 36 x y^{3} + 72 x y^{2} - 44 x y + 8 x + 9 y^{3} - 18 y^{2} + 11 y - 2\right)}{4}\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{6}:v\mapsto v(0,\tfrac{1}{3})\)
\(\displaystyle \phi_{6} = \frac{9 y \left(- 27 x^{3} y^{2} + 45 x^{3} y - 18 x^{3} + 54 x^{2} y^{2} - 90 x^{2} y + 36 x^{2} - 33 x y^{2} + 55 x y - 22 x + 6 y^{2} - 10 y + 4\right)}{4}\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle \phi_{6} = \frac{9 y \left(- 27 x^{3} y^{2} + 45 x^{3} y - 18 x^{3} + 54 x^{2} y^{2} - 90 x^{2} y + 36 x^{2} - 33 x y^{2} + 55 x y - 22 x + 6 y^{2} - 10 y + 4\right)}{4}\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{7}:v\mapsto v(0,\tfrac{2}{3})\)
\(\displaystyle \phi_{7} = \frac{9 y \left(27 x^{3} y^{2} - 36 x^{3} y + 9 x^{3} - 54 x^{2} y^{2} + 72 x^{2} y - 18 x^{2} + 33 x y^{2} - 44 x y + 11 x - 6 y^{2} + 8 y - 2\right)}{4}\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle \phi_{7} = \frac{9 y \left(27 x^{3} y^{2} - 36 x^{3} y + 9 x^{3} - 54 x^{2} y^{2} + 72 x^{2} y - 18 x^{2} + 33 x y^{2} - 44 x y + 11 x - 6 y^{2} + 8 y - 2\right)}{4}\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{8}:v\mapsto v(1,\tfrac{1}{3})\)
\(\displaystyle \phi_{8} = \frac{9 x y \left(27 x^{2} y^{2} - 45 x^{2} y + 18 x^{2} - 27 x y^{2} + 45 x y - 18 x + 6 y^{2} - 10 y + 4\right)}{4}\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle \phi_{8} = \frac{9 x y \left(27 x^{2} y^{2} - 45 x^{2} y + 18 x^{2} - 27 x y^{2} + 45 x y - 18 x + 6 y^{2} - 10 y + 4\right)}{4}\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{9}:v\mapsto v(1,\tfrac{2}{3})\)
\(\displaystyle \phi_{9} = \frac{9 x y \left(- 27 x^{2} y^{2} + 36 x^{2} y - 9 x^{2} + 27 x y^{2} - 36 x y + 9 x - 6 y^{2} + 8 y - 2\right)}{4}\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle \phi_{9} = \frac{9 x y \left(- 27 x^{2} y^{2} + 36 x^{2} y - 9 x^{2} + 27 x y^{2} - 36 x y + 9 x - 6 y^{2} + 8 y - 2\right)}{4}\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{10}:v\mapsto v(\tfrac{1}{3},1)\)
\(\displaystyle \phi_{10} = \frac{9 x y \left(27 x^{2} y^{2} - 27 x^{2} y + 6 x^{2} - 45 x y^{2} + 45 x y - 10 x + 18 y^{2} - 18 y + 4\right)}{4}\)
This DOF is associated with edge 3 of the reference element.
\(\displaystyle \phi_{10} = \frac{9 x y \left(27 x^{2} y^{2} - 27 x^{2} y + 6 x^{2} - 45 x y^{2} + 45 x y - 10 x + 18 y^{2} - 18 y + 4\right)}{4}\)
This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{11}:v\mapsto v(\tfrac{2}{3},1)\)
\(\displaystyle \phi_{11} = \frac{9 x y \left(- 27 x^{2} y^{2} + 27 x^{2} y - 6 x^{2} + 36 x y^{2} - 36 x y + 8 x - 9 y^{2} + 9 y - 2\right)}{4}\)
This DOF is associated with edge 3 of the reference element.
\(\displaystyle \phi_{11} = \frac{9 x y \left(- 27 x^{2} y^{2} + 27 x^{2} y - 6 x^{2} + 36 x y^{2} - 36 x y + 8 x - 9 y^{2} + 9 y - 2\right)}{4}\)
This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{12}:v\mapsto v(\tfrac{1}{3},\tfrac{1}{3})\)
\(\displaystyle \phi_{12} = \frac{81 x y \left(9 x^{2} y^{2} - 15 x^{2} y + 6 x^{2} - 15 x y^{2} + 25 x y - 10 x + 6 y^{2} - 10 y + 4\right)}{4}\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle \phi_{12} = \frac{81 x y \left(9 x^{2} y^{2} - 15 x^{2} y + 6 x^{2} - 15 x y^{2} + 25 x y - 10 x + 6 y^{2} - 10 y + 4\right)}{4}\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{13}:v\mapsto v(\tfrac{2}{3},\tfrac{1}{3})\)
\(\displaystyle \phi_{13} = \frac{81 x y \left(- 9 x^{2} y^{2} + 15 x^{2} y - 6 x^{2} + 12 x y^{2} - 20 x y + 8 x - 3 y^{2} + 5 y - 2\right)}{4}\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle \phi_{13} = \frac{81 x y \left(- 9 x^{2} y^{2} + 15 x^{2} y - 6 x^{2} + 12 x y^{2} - 20 x y + 8 x - 3 y^{2} + 5 y - 2\right)}{4}\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{14}:v\mapsto v(\tfrac{1}{3},\tfrac{2}{3})\)
\(\displaystyle \phi_{14} = \frac{81 x y \left(- 9 x^{2} y^{2} + 12 x^{2} y - 3 x^{2} + 15 x y^{2} - 20 x y + 5 x - 6 y^{2} + 8 y - 2\right)}{4}\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle \phi_{14} = \frac{81 x y \left(- 9 x^{2} y^{2} + 12 x^{2} y - 3 x^{2} + 15 x y^{2} - 20 x y + 5 x - 6 y^{2} + 8 y - 2\right)}{4}\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{15}:v\mapsto v(\tfrac{2}{3},\tfrac{2}{3})\)
\(\displaystyle \phi_{15} = \frac{81 x y \left(9 x^{2} y^{2} - 12 x^{2} y + 3 x^{2} - 12 x y^{2} + 16 x y - 4 x + 3 y^{2} - 4 y + 1\right)}{4}\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle \phi_{15} = \frac{81 x y \left(9 x^{2} y^{2} - 12 x^{2} y + 3 x^{2} - 12 x y^{2} + 16 x y - 4 x + 3 y^{2} - 4 y + 1\right)}{4}\)
This DOF is associated with face 0 of the reference element.