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Bubble enriched Lagrange

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Orders\(1\leqslant k\leqslant 2\)
Reference elementstriangle
Polynomial set\(\mathcal{P}_{k} \oplus \mathcal{Z}^{(14)}_{k+2}\)
↓ Show polynomial set definitions ↓
DOFsOn each vertex: point evaluations
On each edge: point evaluations
On each face: point evaluations
Number of DOFstriangle: \((k+1)^2\) (A000290)
CategoriesScalar-valued elements

Implementations

Symfem"bubble enriched Lagrange"
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Examples

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

\(\displaystyle \phi_{0} = 9 x^{2} y + 9 x y^{2} - 9 x y - x - y + 1\)

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

\(\displaystyle \phi_{1} = x \left(9 x y + 9 y^{2} - 9 y + 1\right)\)

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

\(\displaystyle \phi_{2} = y \left(9 x^{2} + 9 x y - 9 x + 1\right)\)

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

\(\displaystyle \phi_{3} = 27 x y \left(- x - y + 1\right)\)

This DOF is associated with face 0 of the reference element.
  • \(R\) is the reference triangle. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(1\), \(x\), \(x^{2}\), \(y\), \(x y\), \(y^{2}\), \(x y \left(- x - y + 1\right)\), \(x^{2} y \left(- x - y + 1\right)\), \(x y^{2} \left(- x - y + 1\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{8}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0,0)\)

\(\displaystyle \phi_{0} = - 16 x^{3} y - 32 x^{2} y^{2} + 24 x^{2} y + 2 x^{2} - 16 x y^{3} + 24 x y^{2} - 4 x y - 3 x + 2 y^{2} - 3 y + 1\)

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

\(\displaystyle \phi_{1} = x \left(16 x^{2} y + 16 x y^{2} - 24 x y + 2 x - 8 y^{2} + 8 y - 1\right)\)

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

\(\displaystyle \phi_{2} = y \left(16 x^{2} y - 8 x^{2} + 16 x y^{2} - 24 x y + 8 x + 2 y - 1\right)\)

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

\(\displaystyle \phi_{3} = 4 x y \left(8 x^{2} + 16 x y - 10 x + 8 y^{2} - 10 y + 3\right)\)

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

\(\displaystyle \phi_{4} = 4 y \left(- 8 x^{3} - 8 x^{2} y + 14 x^{2} + 6 x y - 7 x - y + 1\right)\)

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

\(\displaystyle \phi_{5} = 4 x \left(- 8 x y^{2} + 6 x y - x - 8 y^{3} + 14 y^{2} - 7 y + 1\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{6}:v\mapsto v(\tfrac{1}{4},\tfrac{1}{4})\)

\(\displaystyle \phi_{6} = 32 x y \left(4 x^{2} + 8 x y - 7 x + 4 y^{2} - 7 y + 3\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{7}:v\mapsto v(\tfrac{1}{4},\tfrac{1}{2})\)

\(\displaystyle \phi_{7} = 32 x y \left(- 4 x y + x - 4 y^{2} + 5 y - 1\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{8}:v\mapsto v(\tfrac{1}{2},\tfrac{1}{4})\)

\(\displaystyle \phi_{8} = 32 x y \left(- 4 x^{2} - 4 x y + 5 x + y - 1\right)\)

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

DefElement stats

Element added02 March 2021
Element last updated21 April 2021