an encyclopedia of finite element definitions

Crouzeix–Raviart

 Abbreviated names CR Orders $$k=1$$ Reference elements triangle, tetrahedron Polynomial set $$\mathcal{P}_{k}$$↓ Show polynomial set definitions ↓ DOFs On each facet: point evaluation at midpoint Number of DOFs triangle: $$3$$tetrahedron: $$4$$ Categories Scalar-valued elements

Implementations

 Symfem string "Crouzeix-Raviart"↓ Show Symfem examples ↓ Basix string "Crouzeix-Raviart"↓ Show Basix examples ↓ UFL string "Crouzeix-Raviart"↓ Show UFL examples ↓

Examples

triangle
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.

References

• Crouzeix, M. and Raviart, P. Conforming and nonconforming finite element methods for solving the stationary Stokes equations, Revue Française d'Automatique, Informatique et Recherche Opérationnelle 3, 33–75, 1973. [DOI: 10.1051/m2an/197307R300331] [BibTeX]