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Rannacher–Turek

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Degrees	\(k=1\) where \(k\) is the Polynomial subdegree
Polynomial subdegree	\(k\)
Polynomial superdegree	\(k+1\)
Lagrange subdegree	\(k-1\)
Lagrange superdegree	\(k+1\)
Reference cells	quadrilateral, hexahedron
Polynomial set	\(\mathcal{P}_{k} \oplus \mathcal{Z}^{(22)}_{k}\) (quadrilateral) \(\mathcal{P}_{k} \oplus \mathcal{Z}^{(23)}_{k}\) (hexahedron) ↓ Show polynomial set definitions ↓↑ Hide polynomial set definitions ↑ \(\mathcal{P}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\sum_{i=1}^dp_i\leqslant k\right\}\) \(\mathcal{Z}^{(22)}_k=\operatorname{span}\left\{(x_1+x_2)(x_1-x_2)\right\}\) \(\mathcal{Z}^{(23)}_k=\operatorname{span}\left\{(x_1+x_2)(x_1-x_2),(x_2+x_3)(x_2-x_3))\right\}\)
DOFs	On each facet: point evaluation at midpoint
Number of DOFs	quadrilateral: \(4\) hexahedron: \(6\)
Mapping	identity
continuity	Discontinuous.
Categories	Scalar-valued elements

Implementations

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Examples

quadrilateral
degree 1

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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	04 June 2025