Trimmed serendipity H(curl)

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

\(\mathcal{Z}^{(40)}_k=\operatorname{span}\left\{\left(\begin{array}{c}y\\-x\end{array}\right)p\middle|p\in\hat{\mathcal{P}}_{k-1}\right\}\)

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

\(\mathcal{Z}^{(41)}_k=\operatorname{span}\left\{\left(\begin{array}{c}y^k\\kxy^{k-1}\end{array}\right),\left(\begin{array}{c}kx^{k-1}y\\x^k\end{array}\right)\right\}\)

\(\mathcal{Z}^{(42)}_k=\operatorname{span}\left\{\boldsymbol{x}\times p\middle|p\in\hat{\mathcal{P}}_{k-1}\right\}\)

\(\mathcal{Z}^{(43)}_k=\operatorname{span}\left\{\nabla p\middle|p\in\{xyz^k,xy^kz,x^kyz\}\cup\{xy^az^{k-a}|a=0,1,...,k\}\cup\{x^ayz^{k-a}|a=0,1,...,k\}\cup\{x^ay^{k-a}z|a=0,1,...,k\}\right\}\)

\(\mathcal{Z}^{(44)}_k=\operatorname{span}\left\{\left(\left(\begin{array}{c}-y\\x\\0\end{array}\right)xy^az^{k-1-a}\right)\middle|a=0,1,...,a-1\right\}\)

\(\mathcal{Z}^{(45)}_k=\operatorname{span}\left\{\left(\left(\begin{array}{c}-z\\0\\x\end{array}\right)yx^az^{k-1-a}\right)\middle|a=0,1,...,a-1\right\}\)

\(\mathcal{Z}^{(46)}_k=\operatorname{span}\left\{\left(\left(\begin{array}{c}-y\\x\\0\end{array}\right)zx^ay^{k-1-a}\right)\middle|a=0,1,...,a-1\right\}\)

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(curl) order 1 on a quadrilateral
element = symfem.create_element("quadrilateral", "TScurl", 1)

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

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

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

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

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