Trimmed serendipity H(div)

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

\(\mathcal{Z}^{(5)}_k=\operatorname{span}\left\{\boldsymbol{x}p\middle|p\in\hat{\mathcal{P}}_{k-1}\right\}\)

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

\(\mathcal{Z}^{(6)}_k=\operatorname{span}\left\{\left(\begin{array}{c}kxy^{k-1}\\-y^k\end{array}\right),\left(\begin{array}{c}-x^k\\kx^{k-1}y\end{array}\right)\right\}\)

\(\mathcal{Z}^{(7)}_k=\operatorname{span}\left\{\operatorname{curl}\left(\left(\begin{array}{c}0\\-z\\y\end{array}\right)xy^az^{k-1-a}\right)\middle|a=0,1,...,a-1\right\}\)

\(\mathcal{Z}^{(8)}_k=\operatorname{span}\left\{\operatorname{curl}\left(\left(\begin{array}{c}-z\\0\\x\end{array}\right)yx^az^{k-1-a}\right)\middle|a=0,1,...,a-1\right\}\)

\(\mathcal{Z}^{(9)}_k=\operatorname{span}\left\{\operatorname{curl}\left(\left(\begin{array}{c}-y\\x\\0\end{array}\right)zx^ay^{k-1-a}\right)\middle|a=0,1,...,a-1\right\}\)

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 trimmed serendipity H(div) order 1 on a quadrilateral
element = symfem.create_element("quadrilateral", "TSdiv", 1)

# Create trimmed serendipity H(div) order 2 on a quadrilateral
element = symfem.create_element("quadrilateral", "TSdiv", 2)

# Create trimmed serendipity H(div) order 3 on a quadrilateral
element = symfem.create_element("quadrilateral", "TSdiv", 3)

# Create trimmed serendipity H(div) order 1 on a hexahedron
element = symfem.create_element("hexahedron", "TSdiv", 1)

# Create trimmed serendipity H(div) order 2 on a hexahedron
element = symfem.create_element("hexahedron", "TSdiv", 2)

# Create trimmed serendipity H(div) order 3 on a hexahedron
element = symfem.create_element("hexahedron", "TSdiv", 3)