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Rannacher–Turek

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Orders\(k=1\)
Reference elementsquadrilateral, hexahedron
Polynomial set\(\mathcal{P}_{k} \oplus \mathcal{Z}^{(1)}_{k}\) (quadrilateral)
\(\mathcal{P}_{k} \oplus \mathcal{Z}^{(2)}_{k}\) (hexahedron)
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DOFsOn each facet: point evaluation at midpoint
Number of DOFsquadrilateral: \(4\)
hexahedron: \(6\)
CategoriesScalar-valued elements

Implementations

Symfem"Rannacher-Turek"
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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

DefElement stats

Element added06 May 2022
Element last updated09 May 2022