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# Rannacher–Turek

 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

 Symfem "Rannacher-Turek"↓ Show Symfem examples ↓

## Examples

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