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Rannacher–Turek
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| Orders | \(k=1\) |
| Reference elements | quadrilateral, hexahedron |
| Polynomial set | \(\mathcal{P}_{k} \oplus \mathcal{Z}^{(1)}_{k}\) (quadrilateral) \(\mathcal{P}_{k} \oplus \mathcal{Z}^{(2)}_{k}\) (hexahedron) ↓ Show polynomial set definitions ↓ |
| DOFs | On each facet: point evaluation at midpoint |
| Number of DOFs | quadrilateral: \(4\) hexahedron: \(6\) |
| Categories | Scalar-valued elements |
Implementations
"Rannacher-Turek"↓ Show Symfem examples ↓ |
Examples
quadrilateralorder 1
- \(R\) 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.
References
- Rannacher, Rolf and Turek, Stefan. Simple nonconforming quadrilateral Stokes element, Numerical methods for partial differential equations 8, 97–111, 1992. [DOI: 10.1002/num.1690080202] [BibTeX]
DefElement stats
| Element added | 06 May 2022 |
| Element last updated | 09 May 2022 |