Brezzi–Douglas–Durán–Fortin

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

\(\mathcal{Z}^{(8)}_k=\operatorname{span}\left\{\nabla\times(0,0,x^{k+1}y), \nabla\times(0,xz^{k+1},0), \nabla\times(y^{k+1}z,0,0)\right\}\)

\(\mathcal{Z}^{(9)}_k=\left\{\nabla\times(0,0,xy^{i+1}z^{k-i}), \nabla\times(0,x^{i+1}y^{k-i}z,0), \nabla\times(x^{k-i}yz^{i+1},0,0)\middle|i=1,...,k\right\}\)

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-Duran-Fortin order 1 on a hexahedron
element = symfem.create_element("hexahedron", "BDDF", 1)

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