an encyclopedia of finite element definitions
Degree 5 Bell 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: \(1\), \(x\), \(x^{2}\), \(x^{3}\), \(x^{4}\), \(y\), \(x y\), \(x^{2} y\), \(x^{3} y\), \(y^{2}\), \(x y^{2}\), \(x^{2} y^{2}\), \(y^{3}\), \(x y^{3}\), \(y^{4}\), \(x^{5} - y^{5}\), \(x^{2} y^{2} \left(x - y\right)\), \(x^{2} \left(- x^{3} + 5 y^{3}\right)\)
- \(\mathcal{L}=\{l_0,...,l_{17}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0,0)\)
\(\displaystyle \phi_{0} = - 6 x^{5} + 15 x^{4} + 30 x^{3} y^{2} - 10 x^{3} + 30 x^{2} y^{3} - 30 x^{2} y^{2} - 6 y^{5} + 15 y^{4} - 10 y^{3} + 1\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \phi_{0} = - 6 x^{5} + 15 x^{4} + 30 x^{3} y^{2} - 10 x^{3} + 30 x^{2} y^{3} - 30 x^{2} y^{2} - 6 y^{5} + 15 y^{4} - 10 y^{3} + 1\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:v\mapsto\frac{\partial}{\partial x}v(0,0)\)
\(\displaystyle \phi_{1} = x \left(- 3 x^{4} + 8 x^{3} + 9 x^{2} y^{2} - 6 x^{2} + 6 x y^{3} - 6 x y^{2} + 2 y^{3} - 3 y^{2} + 1\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \phi_{1} = x \left(- 3 x^{4} + 8 x^{3} + 9 x^{2} y^{2} - 6 x^{2} + 6 x y^{3} - 6 x y^{2} + 2 y^{3} - 3 y^{2} + 1\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{2}:v\mapsto\frac{\partial}{\partial y}v(0,0)\)
\(\displaystyle \phi_{2} = y \left(6 x^{3} y + 2 x^{3} + 9 x^{2} y^{2} - 6 x^{2} y - 3 x^{2} - 3 y^{4} + 8 y^{3} - 6 y^{2} + 1\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \phi_{2} = y \left(6 x^{3} y + 2 x^{3} + 9 x^{2} y^{2} - 6 x^{2} y - 3 x^{2} - 3 y^{4} + 8 y^{3} - 6 y^{2} + 1\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{3}:v\mapsto\frac{\partial^{2}}{\partial x^{2}}v(0,0)\)
\(\displaystyle \phi_{3} = \frac{x^{2} \left(- x^{3} + 3 x^{2} + 3 x y^{2} - 3 x + 2 y^{3} - 3 y^{2} + 1\right)}{2}\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \phi_{3} = \frac{x^{2} \left(- x^{3} + 3 x^{2} + 3 x y^{2} - 3 x + 2 y^{3} - 3 y^{2} + 1\right)}{2}\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{4}:v\mapsto\frac{\partial^{2}}{\partial x\partial y}v(0,0)\)
\(\displaystyle \phi_{4} = x y \left(x^{2} + 2 x y - 2 x + y^{2} - 2 y + 1\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \phi_{4} = x y \left(x^{2} + 2 x y - 2 x + y^{2} - 2 y + 1\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{5}:v\mapsto\frac{\partial^{2}}{\partial y^{2}}v(0,0)\)
\(\displaystyle \phi_{5} = \frac{y^{2} \left(2 x^{3} + 3 x^{2} y - 3 x^{2} - y^{3} + 3 y^{2} - 3 y + 1\right)}{2}\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \phi_{5} = \frac{y^{2} \left(2 x^{3} + 3 x^{2} y - 3 x^{2} - y^{3} + 3 y^{2} - 3 y + 1\right)}{2}\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{6}:v\mapsto v(1,0)\)
\(\displaystyle \phi_{6} = x^{2} \left(6 x^{3} - 15 x^{2} - 15 x y^{2} + 10 x - 15 y^{3} + 15 y^{2}\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \phi_{6} = x^{2} \left(6 x^{3} - 15 x^{2} - 15 x y^{2} + 10 x - 15 y^{3} + 15 y^{2}\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{7}:v\mapsto\frac{\partial}{\partial x}v(1,0)\)
\(\displaystyle \phi_{7} = \frac{x^{2} \left(- 6 x^{3} + 14 x^{2} + 15 x y^{2} - 8 x + 15 y^{3} - 15 y^{2}\right)}{2}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \phi_{7} = \frac{x^{2} \left(- 6 x^{3} + 14 x^{2} + 15 x y^{2} - 8 x + 15 y^{3} - 15 y^{2}\right)}{2}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{8}:v\mapsto\frac{\partial}{\partial y}v(1,0)\)
\(\displaystyle \phi_{8} = \frac{x^{2} y \left(3 x y - 4 x - 3 y^{2} - 3 y + 6\right)}{2}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \phi_{8} = \frac{x^{2} y \left(3 x y - 4 x - 3 y^{2} - 3 y + 6\right)}{2}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{9}:v\mapsto\frac{\partial^{2}}{\partial x^{2}}v(1,0)\)
\(\displaystyle \phi_{9} = \frac{x^{2} \left(2 x^{3} - 4 x^{2} - 5 x y^{2} + 2 x - 5 y^{3} + 5 y^{2}\right)}{4}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \phi_{9} = \frac{x^{2} \left(2 x^{3} - 4 x^{2} - 5 x y^{2} + 2 x - 5 y^{3} + 5 y^{2}\right)}{4}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{10}:v\mapsto\frac{\partial^{2}}{\partial x\partial y}v(1,0)\)
\(\displaystyle \phi_{10} = \frac{x^{2} y \left(- x y + 2 x + y^{2} + y - 2\right)}{2}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \phi_{10} = \frac{x^{2} y \left(- x y + 2 x + y^{2} + y - 2\right)}{2}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{11}:v\mapsto\frac{\partial^{2}}{\partial y^{2}}v(1,0)\)
\(\displaystyle \phi_{11} = \frac{x^{2} y^{2} \left(x - y + 1\right)}{4}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \phi_{11} = \frac{x^{2} y^{2} \left(x - y + 1\right)}{4}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{12}:v\mapsto v(0,1)\)
\(\displaystyle \phi_{12} = y^{2} \left(- 15 x^{3} - 15 x^{2} y + 15 x^{2} + 6 y^{3} - 15 y^{2} + 10 y\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \phi_{12} = y^{2} \left(- 15 x^{3} - 15 x^{2} y + 15 x^{2} + 6 y^{3} - 15 y^{2} + 10 y\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{13}:v\mapsto\frac{\partial}{\partial x}v(0,1)\)
\(\displaystyle \phi_{13} = \frac{x y^{2} \left(- 3 x^{2} + 3 x y - 3 x - 4 y + 6\right)}{2}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \phi_{13} = \frac{x y^{2} \left(- 3 x^{2} + 3 x y - 3 x - 4 y + 6\right)}{2}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{14}:v\mapsto\frac{\partial}{\partial y}v(0,1)\)
\(\displaystyle \phi_{14} = \frac{y^{2} \left(15 x^{3} + 15 x^{2} y - 15 x^{2} - 6 y^{3} + 14 y^{2} - 8 y\right)}{2}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \phi_{14} = \frac{y^{2} \left(15 x^{3} + 15 x^{2} y - 15 x^{2} - 6 y^{3} + 14 y^{2} - 8 y\right)}{2}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{15}:v\mapsto\frac{\partial^{2}}{\partial x^{2}}v(0,1)\)
\(\displaystyle \phi_{15} = \frac{x^{2} y^{2} \left(- x + y + 1\right)}{4}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \phi_{15} = \frac{x^{2} y^{2} \left(- x + y + 1\right)}{4}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{16}:v\mapsto\frac{\partial^{2}}{\partial x\partial y}v(0,1)\)
\(\displaystyle \phi_{16} = \frac{x y^{2} \left(x^{2} - x y + x + 2 y - 2\right)}{2}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \phi_{16} = \frac{x y^{2} \left(x^{2} - x y + x + 2 y - 2\right)}{2}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{17}:v\mapsto\frac{\partial^{2}}{\partial y^{2}}v(0,1)\)
\(\displaystyle \phi_{17} = \frac{y^{2} \left(- 5 x^{3} - 5 x^{2} y + 5 x^{2} + 2 y^{3} - 4 y^{2} + 2 y\right)}{4}\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \phi_{17} = \frac{y^{2} \left(- 5 x^{3} - 5 x^{2} y + 5 x^{2} + 2 y^{3} - 4 y^{2} + 2 y\right)}{4}\)
This DOF is associated with vertex 2 of the reference element.