an encyclopedia of finite element definitions


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Abbreviated namesCR
Reference elementstriangle, tetrahedron
Polynomial set\(\mathcal{P}_{k}\)
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DOFsOn each facet: point evaluation at midpoint
Number of DOFstriangle: \(3\)
tetrahedron: \(4\)
CategoriesScalar-valued elements


Symfem string"Crouzeix-Raviart"
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Basix string"Crouzeix-Raviart"
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UFL string"Crouzeix-Raviart"
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order 1
  • \(R\) is the reference triangle. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(1\), \(x\), \(y\)
  • \(\mathcal{L}=\{l_0,...,l_{2}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(\tfrac{1}{2},\tfrac{1}{2})\)

\(\displaystyle \phi_{0} = 2 x + 2 y - 1\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{1}:v\mapsto v(0,\tfrac{1}{2})\)

\(\displaystyle \phi_{1} = 1 - 2 x\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{2}:v\mapsto v(\tfrac{1}{2},0)\)

\(\displaystyle \phi_{2} = 1 - 2 y\)

This DOF is associated with edge 2 of the reference element.