an encyclopedia of finite element definitions

Bubble

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Ordersinterval: \(2\leqslant k\)
triangle: \(3\leqslant k\)
tetrahedron: \(4\leqslant k\)
Reference elementsinterval, triangle, tetrahedron
Polynomial set\(\mathcal{Z}^{(6)}_{k}\)
↓ Show polynomial set definitions ↓
DOFsOn the interior of the reference element: point evaluations
Number of DOFsinterval: \(k-1\) (A000027)
triangle: \((k-2)(k-1)/2\) (A000217)
tetrahedron: \((k-3)(k-2)(k-1)/6\) (A000292)
CategoriesScalar-valued elements

Implementations

Symfem string"bubble"
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Basix string"Bubble"
↓ Show Basix examples ↓
UFL string"Bubble"
↓ Show UFL examples ↓

Examples

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

\(\displaystyle \phi_{0} = 4 x \left(1 - x\right)\)

This DOF is associated with edge 0 of the reference element.
  • \(R\) is the reference interval. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(x \left(1 - x\right)\), \(x^{2} \left(1 - x\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{1}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(\tfrac{1}{3})\)

\(\displaystyle \phi_{0} = \frac{9 x \left(3 x^{2} - 5 x + 2\right)}{2}\)

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

\(\displaystyle \phi_{1} = \frac{9 x \left(- 3 x^{2} + 4 x - 1\right)}{2}\)

This DOF is associated with edge 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: \(x y \left(- x - y + 1\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{0}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(\tfrac{1}{3},\tfrac{1}{3})\)

\(\displaystyle \phi_{0} = 27 x y \left(- x - y + 1\right)\)

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: \(x y \left(- x - y + 1\right)\), \(x^{2} y \left(- x - y + 1\right)\), \(x y^{2} \left(- x - y + 1\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{2}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(\tfrac{1}{4},\tfrac{1}{4})\)

\(\displaystyle \phi_{0} = 32 x y \left(4 x^{2} + 8 x y - 7 x + 4 y^{2} - 7 y + 3\right)\)

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

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

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

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

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

References