Serendipity H(div)

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

\(\mathcal{Z}^{(29)}_k=\operatorname{span}\left\{\left(\begin{array}{c}x^{k+1}\\(k+1)x^ky\end{array}\right),\left(\begin{array}{c}(k+1)xy^k\\y^{k+1}\end{array}\right)\right\}\)

\(\mathcal{Z}^{(30)}_k=\left\{\nabla\times\boldsymbol{p}\middle|\boldsymbol{p}\in\mathcal{A}_{k}\right\}\)

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

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(div) order 1 on a quadrilateral
element = symfem.create_element("quadrilateral", "Sdiv", 1)

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

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

# Create serendipity H(div) order 2 on a hexahedron
element = symfem.create_element("hexahedron", "Sdiv", 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(div) order 1 on a quadrilateral
element = ufl_legacy.FiniteElement("BDMCF", "quadrilateral", 1)

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

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

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