Q H(div)

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

\(\mathcal{Z}^{(23)}_k=\left\{\boldsymbol{q}\in\hat{\mathcal{Q}}_{k}\middle|\boldsymbol{q}(\boldsymbol{x})\cdot x_i\boldsymbol{e}_j\in\begin{cases}\hat{\mathcal{Q}}_{k+1}&i=j\\\hat{\mathcal{Q}}_{k}&i\not=j\end{cases}\text{ for }i,j=1,\dots,d\right\}\)

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

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 Q H(div) order 1 on a quadrilateral
element = symfem.create_element("quadrilateral", "Qdiv", 1)

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

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

# Create Q H(div) order 2 on a hexahedron
element = symfem.create_element("hexahedron", "Qdiv", 2)

Before trying this example, you must install (legacy) UFL:

pip3 install git+https://github.com/FEniCS/ufl.git@ufl_legacy

This element can then be created with the following lines of Python:

import ufl_legacy

# Create Q H(div) order 1 on a quadrilateral
element = ufl_legacy.FiniteElement("RTCF", "quadrilateral", 1)

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

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

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