Brezzi–Douglas–Fortin–Marini

\(\mathcal{Z}^{(10)}_k=\left\{\boldsymbol{p}\in\mathcal{P}_{k}\middle|\boldsymbol{p}\cdot\boldsymbol{n}\in\mathcal{P}_{k-1}\text{ on each facet}\right\}\)

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

Before trying this example, you must install 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 Brezzi-Douglas-Fortin-Marini order 2
element = FIAT.BrezziDouglasFortinMarini(FIAT.ufc_cell("triangle"), 2)

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 Brezzi-Douglas-Fortin-Marini order 1 on a triangle
element = symfem.create_element("triangle", "BDFM", 1)

# Create Brezzi-Douglas-Fortin-Marini order 2 on a triangle
element = symfem.create_element("triangle", "BDFM", 2)

# Create Brezzi-Douglas-Fortin-Marini order 1 on a quadrilateral
element = symfem.create_element("quadrilateral", "BDFM", 1)

# Create Brezzi-Douglas-Fortin-Marini order 2 on a quadrilateral
element = symfem.create_element("quadrilateral", "BDFM", 2)

# Create Brezzi-Douglas-Fortin-Marini order 2 on a tetrahedron
element = symfem.create_element("tetrahedron", "BDFM", 2)

# Create Brezzi-Douglas-Fortin-Marini order 2 on a hexahedron
element = symfem.create_element("hexahedron", "BDFM", 2)

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 Brezzi-Douglas-Fortin-Marini order 1 on a triangle
element = ufl_legacy.FiniteElement("BDFM", "triangle", 1)

# Create Brezzi-Douglas-Fortin-Marini order 2 on a triangle
element = ufl_legacy.FiniteElement("BDFM", "triangle", 2)

# Create Brezzi-Douglas-Fortin-Marini order 2 on a tetrahedron
element = ufl_legacy.FiniteElement("BDFM", "tetrahedron", 2)