an encyclopedia of finite element definitions

# Crouzeix–Falk

 Abbreviated names FS Orders $$k=3$$ Reference elements triangle Polynomial set $$\mathcal{P}_{k}$$↓ Show polynomial set definitions ↓ DOFs On each facet: point evaluation at two points (but only one point on one of the edges) On the interior of the reference element: point evaluation at midpoint Number of DOFs triangle: $$10$$ Categories Scalar-valued elements

## Implementations

 Symfem "Crouzeix-Falk"↓ Show Symfem examples ↓

## Examples

triangle
order 3
• $$R$$ is the reference triangle. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$1$$, $$x$$, $$x^{2}$$, $$x^{3}$$, $$y$$, $$x y$$, $$x^{2} y$$, $$y^{2}$$, $$x y^{2}$$, $$y^{3}$$
• $$\mathcal{L}=\{l_0,...,l_{9}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto v(\tfrac{3}{4},\tfrac{1}{4})$$

$$\displaystyle \phi_{0} = \frac{64 x^{3}}{3} + 42 x^{2} y - 32 x^{2} + \frac{94 x y^{2}}{3} - \frac{134 x y}{3} + \frac{44 x}{3} + \frac{64 y^{3}}{3} - 32 y^{2} + \frac{44 y}{3} - 2$$

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

$$\displaystyle \phi_{1} = - 32 x^{3} - 43 x^{2} y + 48 x^{2} - 43 x y^{2} + 59 x y - 22 x - 32 y^{3} + 48 y^{2} - 22 y + 3$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{2}:v\mapsto v(\tfrac{1}{4},\tfrac{3}{4})$$

$$\displaystyle \phi_{2} = \frac{64 x^{3}}{3} + \frac{94 x^{2} y}{3} - 32 x^{2} + 42 x y^{2} - \frac{134 x y}{3} + \frac{44 x}{3} + \frac{64 y^{3}}{3} - 32 y^{2} + \frac{44 y}{3} - 2$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{3}:v\mapsto v(0,\tfrac{1}{4})$$

$$\displaystyle \phi_{3} = - \frac{64 x^{3}}{3} - 22 x^{2} y + 32 x^{2} - \frac{34 x y^{2}}{3} + \frac{74 x y}{3} - \frac{44 x}{3} + \frac{32 y^{3}}{3} - 8 y^{2} - \frac{8 y}{3} + 2$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{4}:v\mapsto v(0,\tfrac{1}{2})$$

$$\displaystyle \phi_{4} = 32 x^{3} + 53 x^{2} y - 48 x^{2} + 53 x y^{2} - 69 x y + 22 x - 16 y^{2} + 16 y - 3$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{5}:v\mapsto v(0,\tfrac{3}{4})$$

$$\displaystyle \phi_{5} = - \frac{64 x^{3}}{3} - \frac{98 x^{2} y}{3} + 32 x^{2} - \frac{130 x y^{2}}{3} + 46 x y - \frac{44 x}{3} - \frac{32 y^{3}}{3} + 24 y^{2} - \frac{40 y}{3} + 2$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{6}:v\mapsto v(\tfrac{1}{4},0)$$

$$\displaystyle \phi_{6} = \frac{32 x^{3}}{3} - \frac{34 x^{2} y}{3} - 8 x^{2} - 22 x y^{2} + \frac{74 x y}{3} - \frac{8 x}{3} - \frac{64 y^{3}}{3} + 32 y^{2} - \frac{44 y}{3} + 2$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{7}:v\mapsto v(\tfrac{1}{2},0)$$

$$\displaystyle \phi_{7} = 53 x^{2} y - 16 x^{2} + 53 x y^{2} - 69 x y + 16 x + 32 y^{3} - 48 y^{2} + 22 y - 3$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{8}:v\mapsto v(\tfrac{3}{4},0)$$

$$\displaystyle \phi_{8} = - \frac{32 x^{3}}{3} - \frac{130 x^{2} y}{3} + 24 x^{2} - \frac{98 x y^{2}}{3} + 46 x y - \frac{40 x}{3} - \frac{64 y^{3}}{3} + 32 y^{2} - \frac{44 y}{3} + 2$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{9}:v\mapsto v(\tfrac{1}{3},\tfrac{1}{3})$$

$$\displaystyle \phi_{9} = 27 x y \left(- x - y + 1\right)$$

This DOF is associated with face 0 of the reference element.

## References

• Crouzeix, M. and Falk, R. S. Nonconforming finite elements for the Stokes problem, Mathematics of Computation 52, 437–456, 1989. [DOI: 10.2307/2008475] [BibTeX]

## DefElement stats

 Element added 06 March 2021 Element last updated 09 November 2021