Bubble

\(\mathcal{Z}^{(11)}_k=\left\{p\in \mathcal{P}_{k}\middle|p=0\text{ on the boundary}\right\}\)

\(\mathcal{P}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle|\sum_{i=1}^dp_i\leqslant k\right\}\)

Before trying this example, you must install Basix:

pip3 install git+https://github.com/FEniCS/basix.git

This element can then be created with the following lines of Python:

import basix

# Create bubble order 2 on a interval
element = basix.create_element(basix.ElementFamily.bubble, basix.CellType.interval, 2)

# Create bubble order 3 on a interval
element = basix.create_element(basix.ElementFamily.bubble, basix.CellType.interval, 3)

# Create bubble order 3 on a triangle
element = basix.create_element(basix.ElementFamily.bubble, basix.CellType.triangle, 3)

# Create bubble order 4 on a triangle
element = basix.create_element(basix.ElementFamily.bubble, basix.CellType.triangle, 4)

Before trying this example, you must install Basix.UFL:

pip3 install git+https://github.com/FEniCS/basix.git git+https://github.com/FEniCS/ufl.git

This element can then be created with the following lines of Python:

import basix
import basix.ufl

# Create bubble order 2 on a interval
element = basix.ufl.element(basix.ElementFamily.bubble, basix.CellType.interval, 2)

# Create bubble order 3 on a interval
element = basix.ufl.element(basix.ElementFamily.bubble, basix.CellType.interval, 3)

# Create bubble order 3 on a triangle
element = basix.ufl.element(basix.ElementFamily.bubble, basix.CellType.triangle, 3)

# Create bubble order 4 on a triangle
element = basix.ufl.element(basix.ElementFamily.bubble, basix.CellType.triangle, 4)

Before trying 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 order 2
element = FIAT.Bubble(FIAT.ufc_cell("interval"), 2)

# Create bubble order 3
element = FIAT.Bubble(FIAT.ufc_cell("interval"), 3)

# Create bubble order 3
element = FIAT.Bubble(FIAT.ufc_cell("triangle"), 3)

# Create bubble order 4
element = FIAT.Bubble(FIAT.ufc_cell("triangle"), 4)

Before trying 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 order 2 on a interval
element = symfem.create_element("interval", "bubble", 2)

# Create bubble order 3 on a interval
element = symfem.create_element("interval", "bubble", 3)

# Create bubble order 3 on a triangle
element = symfem.create_element("triangle", "bubble", 3)

# Create bubble order 4 on a triangle
element = symfem.create_element("triangle", "bubble", 4)

Before trying this example, you must install (legacy) UFL:

pip3 install fenics-ufl-legacy

This element can then be created with the following lines of Python:

import ufl_legacy

# Create bubble order 2 on a interval
element = ufl_legacy.FiniteElement("Bubble", "interval", 2)

# Create bubble order 3 on a interval
element = ufl_legacy.FiniteElement("Bubble", "interval", 3)

# Create bubble order 3 on a triangle
element = ufl_legacy.FiniteElement("Bubble", "triangle", 3)

# Create bubble order 4 on a triangle
element = ufl_legacy.FiniteElement("Bubble", "triangle", 4)