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 ↓↑ Hide polynomial set definitions ↑\(\mathcal{P}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle|\sum_{i=1}^dp_i\leqslant k\right\}\)
\(\mathcal{Z}^{(1)}_k=\operatorname{span}\left\{(x_1+x_2)(x_1-x_2)\right\}\)
\(\mathcal{Z}^{(2)}_k=\operatorname{span}\left\{(x_1+x_2)(x_1-x_2),(x_2+x_3)(x_2-x_3))\right\}\) |
DOFs | On each facet: point evaluation at midpoint |
Number of DOFs | quadrilateral: \(4\) hexahedron: \(6\) |
Categories | Scalar-valued elements |
Implementations
Symfem | "Rannacher-Turek" ↓ Show Symfem examples ↓↑ Hide Symfem examples ↑Before trying this example, you must install Symfem: pip3 install symfem This element can then be created with the following lines of Python: import symfem
# Create Rannacher-Turek order 1 on a quadrilateral element = symfem.create_element("quadrilateral", "Rannacher-Turek", 1) |
Examples
quadrilateral
order 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.
\(\displaystyle l_{1}:v\mapsto v(0,\tfrac{1}{2})\)
\(\displaystyle \phi_{1} = x^{2} - 2 x - y^{2} + y + \frac{3}{4}\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{2}:v\mapsto v(1,\tfrac{1}{2})\)
\(\displaystyle \phi_{2} = x^{2} - y^{2} + y - \frac{1}{4}\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{3}:v\mapsto v(\tfrac{1}{2},1)\)
\(\displaystyle \phi_{3} = - x^{2} + x + y^{2} - \frac{1}{4}\)
This DOF is associated with edge 3 of the reference element.
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 |