an encyclopedia of finite element definitions
Degree 4 bubble 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: \(x y \left(x^{2} + 2 x y - 2 x + y^{2} - 2 y + 1\right)\), \(x^{2} y \left(- x - y + 1\right)\), \(x y^{2} \left(- x - y + 1\right)\)
- \(\mathcal{L}=\{l_0,...,l_{2}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(\tfrac{1}{4},\tfrac{1}{4})\)
\(\displaystyle \phi_{0} = 32 x y \left(4 x^{2} + 8 x y - 7 x + 4 y^{2} - 7 y + 3\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle \phi_{0} = 32 x y \left(4 x^{2} + 8 x y - 7 x + 4 y^{2} - 7 y + 3\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{1}:v\mapsto v(\tfrac{1}{2},\tfrac{1}{4})\)
\(\displaystyle \phi_{1} = 32 x y \left(- 4 x^{2} - 4 x y + 5 x + y - 1\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle \phi_{1} = 32 x y \left(- 4 x^{2} - 4 x y + 5 x + y - 1\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{2}:v\mapsto v(\tfrac{1}{4},\tfrac{1}{2})\)
\(\displaystyle \phi_{2} = 32 x y \left(- 4 x y + x - 4 y^{2} + 5 y - 1\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle \phi_{2} = 32 x y \left(- 4 x y + x - 4 y^{2} + 5 y - 1\right)\)
This DOF is associated with face 0 of the reference element.