an encyclopedia of finite element definitions
Degree 3 transition 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: \(1\), \(x\), \(y\), \(x y \left(- x - y + 1\right)\)
- \(\mathcal{L}=\{l_0,...,l_{3}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0,0)\)
\(\displaystyle \phi_{0} = 9 x^{2} y + 9 x y^{2} - 9 x y - x - y + 1\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \phi_{0} = 9 x^{2} y + 9 x y^{2} - 9 x y - x - y + 1\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:v\mapsto v(1,0)\)
\(\displaystyle \phi_{1} = x \left(9 x y + 9 y^{2} - 9 y + 1\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \phi_{1} = x \left(9 x y + 9 y^{2} - 9 y + 1\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{2}:v\mapsto v(0,1)\)
\(\displaystyle \phi_{2} = y \left(9 x^{2} + 9 x y - 9 x + 1\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \phi_{2} = y \left(9 x^{2} + 9 x y - 9 x + 1\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{3}:v\mapsto v(\tfrac{1}{3},\tfrac{1}{3})\)
\(\displaystyle \phi_{3} = 27 x y \left(- x - y + 1\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle \phi_{3} = 27 x y \left(- x - y + 1\right)\)
This DOF is associated with face 0 of the reference element.