Bernardi–Raugel

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

\(\mathcal{Z}^{(4)}_k=\left\{\left(\begin{array}{c}xy\\xy\end{array}\right),\left(\begin{array}{c}y(1-x-y)\\0\end{array}\right),\left(\begin{array}{c}0\\x(1-x-y)\end{array}\right)\right\}\)

\(\mathcal{Z}^{(5)}_k=\left\{\left(\begin{array}{c}xyz\\xyz\\xyz\end{array}\right),\left(\begin{array}{c}yz(1-x-y-z)\\0\\0\end{array}\right),\left(\begin{array}{c}0\\xz(1-x-y-z)\\0\end{array}\right),\left(\begin{array}{c}0\\0\\xy(1-x-y-z)\end{array}\right)\right\}\)

\(\mathcal{Z}^{(6)}_k=\left\{\left(\begin{array}{c}xyz\\xyz\\xyz\end{array}\right),\left(\begin{array}{c}yz(1-x-y-z)\\0\\0\end{array}\right),\left(\begin{array}{c}0\\xz(1-x-y-z)\\0\end{array}\right),\left(\begin{array}{c}0\\0\\xy(1-x-y-z)\end{array}\right),\left(\begin{array}{c}xyz(1-x-y-z)\\0\\0\end{array}\right),\left(\begin{array}{c}0\\xyz(1-x-y-z)\\0\end{array}\right),\left(\begin{array}{c}0\\0\\xyz(1-x-y-z)\end{array}\right)\right\}\)

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 Bernardi-Raugel order 1 on a triangle
element = symfem.create_element("triangle", "Bernardi-Raugel", 1)

# Create Bernardi-Raugel order 1 on a tetrahedron
element = symfem.create_element("tetrahedron", "Bernardi-Raugel", 1)

# Create Bernardi-Raugel order 2 on a tetrahedron
element = symfem.create_element("tetrahedron", "Bernardi-Raugel", 2)