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Morley

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Orders\(k=2\)
Reference elementstriangle
Polynomial set\(\mathcal{P}_{k}\)
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DOFsOn each vertex: point evaluations
On each edge: point evaluations of normal derivatives at midpoints
Number of DOFstriangle: \(6\)
CategoriesScalar-valued elements

Implementations

Symfem string"Morley"
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UFL string"Morley"
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Examples

triangle
order 2
  • \(R\) is the reference triangle. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(1\), \(x\), \(x^{2}\), \(y\), \(x y\), \(y^{2}\)
  • \(\mathcal{L}=\{l_0,...,l_{5}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0,0)\)

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

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

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

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

\(\displaystyle \phi_{2} = - \frac{x^{2}}{2} - x y + \frac{x}{2} + \frac{y^{2}}{2} + \frac{y}{2}\)

This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{3}:v\mapsto\nabla{v}(\tfrac{1}{2},\tfrac{1}{2})\cdot\hat{\boldsymbol{n}}_{0}\)

\(\displaystyle \phi_{3} = \frac{\sqrt{2} \left(- x^{2} - 2 x y + x - y^{2} + y\right)}{2}\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{4}:v\mapsto\nabla{v}(0,\tfrac{1}{2})\cdot\hat{\boldsymbol{n}}_{1}\)

\(\displaystyle \phi_{4} = x \left(x - 1\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{5}:v\mapsto\nabla{v}(\tfrac{1}{2},0)\cdot\hat{\boldsymbol{n}}_{2}\)

\(\displaystyle \phi_{5} = y \left(1 - y\right)\)

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

References