an encyclopedia of finite element definitions

# Bubble

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 Orders interval: $$2\leqslant k$$ triangle: $$3\leqslant k$$ tetrahedron: $$4\leqslant k$$ Reference elements interval, triangle, tetrahedron Polynomial set $$\mathcal{Z}^{(3)}_{k}$$↓ Show polynomial set definitions ↓ DOFs On the interior of the reference element: point evaluations Number of DOFs interval: $$k-1$$ (A000027)triangle: $$(k-2)(k-1)/2$$ (A000217)tetrahedron: $$(k-3)(k-2)(k-1)/6$$ (A000292) Categories Scalar-valued elements

## Implementations

 Basix basix.ElementFamily.bubble↓ Show Basix examples ↓ Symfem "bubble"↓ Show Symfem examples ↓ UFL "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: $$x \left(1 - x\right)$$, $$x^{2} \cdot \left(1 - x\right)$$
• $$\mathcal{L}=\{l_0,...,l_{1}\}$$
• 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

• Kirby, Robert C., Logg, Anders, Rognes, Marie E., and Terrel, Andy R. Common and unusual finite elements, in Automated solution of differential equations by the finite element method (eds: Logg, Anders, Mardal, Kent-Andre, and Wells, Garth N.), 2012. [DOI: 10.1007/978-3-642-23099-8_3] [BibTeX]

## DefElement stats

 Element added 10 January 2021 Element last updated 10 February 2022