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Crouzeix–Falk

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Abbreviated namesFS
Orders\(k=3\)
Reference elementstriangle
Polynomial set\(\mathcal{P}_{k}\)
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DOFsOn 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 DOFstriangle: \(10\)
CategoriesScalar-valued elements

Implementations

Symfem"Crouzeix-Falk"
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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

DefElement stats

Element added06 March 2021
Element last updated09 November 2021