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