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