an encyclopedia of finite element definitions

Bubble

Click here to read what the information on this page means.

Degrees	interval: \(2\leqslant k\) triangle: \(3\leqslant k\) tetrahedron: \(4\leqslant k\) where \(k\) is the Polynomial superdegree
Polynomial subdegree	\(-1\)
Polynomial superdegree	interval: \(k\) triangle: \(k\) tetrahedron: \(k\) quadrilateral: \(dk\) hexahedron: \(dk\)
Reference elements	interval, triangle, tetrahedron
Polynomial set	\(\mathcal{Z}^{(11)}_{k}\) ↓ Show polynomial set definitions ↓↑ Hide polynomial set definitions ↑ \(\mathcal{Z}^{(11)}_k=\left\{p\in \mathcal{P}_{k}\middle\|p=0\text{ on the boundary}\right\}\) \(\mathcal{P}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\sum_{i=1}^dp_i\leqslant k\right\}\)
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)
Mapping	identity
continuity	Function values are continuous.
Categories	Scalar-valued elements

Implementations

This element is implemented in Basix , Basix.UFL , FIAT , Symfem , and (legacy) UFL.↓ Show implementation detail ↓

↑ Hide implementation detail ↑

Basix	`basix.ElementFamily.bubble` ↓ Show Basix examples ↓↑ Hide Basix examples ↑ Before running this example, you must install Basix: pip3 install fenics-basix This element can then be created with the following lines of Python: import basix # Create bubble degree 2 on a interval element = basix.create_element(basix.ElementFamily.bubble, basix.CellType.interval, 2) # Create bubble degree 3 on a interval element = basix.create_element(basix.ElementFamily.bubble, basix.CellType.interval, 3) # Create bubble degree 3 on a triangle element = basix.create_element(basix.ElementFamily.bubble, basix.CellType.triangle, 3) # Create bubble degree 4 on a triangle element = basix.create_element(basix.ElementFamily.bubble, basix.CellType.triangle, 4) This implementation is correct for all the examples below.
Basix.UFL	`basix.ElementFamily.bubble` ↓ Show Basix.UFL examples ↓↑ Hide Basix.UFL examples ↑ Before running this example, you must install Basix.UFL: pip3 install fenics-basix fenics-ufl This element can then be created with the following lines of Python: import basix import basix.ufl # Create bubble degree 2 on a interval element = basix.ufl.element(basix.ElementFamily.bubble, basix.CellType.interval, 2) # Create bubble degree 3 on a interval element = basix.ufl.element(basix.ElementFamily.bubble, basix.CellType.interval, 3) # Create bubble degree 3 on a triangle element = basix.ufl.element(basix.ElementFamily.bubble, basix.CellType.triangle, 3) # Create bubble degree 4 on a triangle element = basix.ufl.element(basix.ElementFamily.bubble, basix.CellType.triangle, 4) This implementation is correct for all the examples below.
FIAT	`FIAT.Bubble` ↓ Show FIAT examples ↓↑ Hide FIAT examples ↑ Before running this example, you must install FIAT: pip3 install git+https://github.com/firedrakeproject/fiat.git This element can then be created with the following lines of Python: import FIAT # Create bubble degree 2 element = FIAT.Bubble(FIAT.ufc_cell("interval"), 2) # Create bubble degree 3 element = FIAT.Bubble(FIAT.ufc_cell("interval"), 3) # Create bubble degree 3 element = FIAT.Bubble(FIAT.ufc_cell("triangle"), 3) # Create bubble degree 4 element = FIAT.Bubble(FIAT.ufc_cell("triangle"), 4) This implementation is correct for all the examples below.
Symfem	`"bubble"` ↓ Show Symfem examples ↓↑ Hide Symfem examples ↑ Before running this example, you must install Symfem: pip3 install symfem This element can then be created with the following lines of Python: import symfem # Create bubble degree 2 on a interval element = symfem.create_element("interval", "bubble", 2) # Create bubble degree 3 on a interval element = symfem.create_element("interval", "bubble", 3) # Create bubble degree 3 on a triangle element = symfem.create_element("triangle", "bubble", 3) # Create bubble degree 4 on a triangle element = symfem.create_element("triangle", "bubble", 4) This implementation is used to compute the examples below and verify other implementations.
(legacy) UFL	`"Bubble"` ↓ Show (legacy) UFL examples ↓↑ Hide (legacy) UFL examples ↑ Before running this example, you must install (legacy) UFL: pip3 install setuptools pip3 install fenics-ufl-legacy This element can then be created with the following lines of Python: import ufl_legacy # Create bubble degree 2 on a interval element = ufl_legacy.FiniteElement("Bubble", "interval", 2) # Create bubble degree 3 on a interval element = ufl_legacy.FiniteElement("Bubble", "interval", 3) # Create bubble degree 3 on a triangle element = ufl_legacy.FiniteElement("Bubble", "triangle", 3) # Create bubble degree 4 on a triangle element = ufl_legacy.FiniteElement("Bubble", "triangle", 4)

Examples

interval degree 2	(click to view basis functions)
interval degree 3	(click to view basis functions)
triangle degree 3	(click to view basis functions)
triangle degree 4	(click to view basis functions)

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	27 September 2024