Serendipity H(curl)

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

\(\mathcal{Z}^{(27)}_k=\operatorname{span}\left\{\left(\begin{array}{c}(k+1)x^ky\\-x^{k+1}\end{array}\right),\left(\begin{array}{c}y^{k+1}\\-(k+1)xy^k\end{array}\right)\right\}\)

\(\mathcal{A}_k=\operatorname{span}\bigcup_{p\in\mathcal{P}_k(\mathbb{R}^{d-1})}\left\{\left(\begin{array}{c}0\\xzp(y,z)\\-xyp(y,z)\end{array}\right),\left(\begin{array}{c}yzp(x,z)\\0\\-xyp(x,z)\end{array}\right),\left(\begin{array}{c}yzp(x,y)\\-xzp(x,y)\\0\end{array}\right)\right\}\)

\(\mathcal{Z}^{(28)}_k=\left\{\nabla p\middle|p\in\mathcal{X}_{k+1}\right\}\)

\(\mathcal{X}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle|k<\sum_{i=1}^da_i\leqslant k+\#\left\{i\in\{1,\dots,d\}\middle| a_i=1\right\}\right\}\)

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 serendipity H(curl) order 1 on a quadrilateral
element = symfem.create_element("quadrilateral", "Scurl", 1)

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

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

# Create serendipity H(curl) order 2 on a hexahedron
element = symfem.create_element("hexahedron", "Scurl", 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 serendipity H(curl) order 1 on a quadrilateral
element = ufl_legacy.FiniteElement("BDMCE", "quadrilateral", 1)

# Create serendipity H(curl) order 2 on a quadrilateral
element = ufl_legacy.FiniteElement("BDMCE", "quadrilateral", 2)

# Create serendipity H(curl) order 1 on a hexahedron
element = ufl_legacy.FiniteElement("AAE", "hexahedron", 1)

# Create serendipity H(curl) order 2 on a hexahedron
element = ufl_legacy.FiniteElement("AAE", "hexahedron", 2)