Arnold–Winther

\(\mathcal{Z}^{(2)}_k=\left\{\mathbf{M}\in\mathcal{P}_{k}^{d\times d}\middle|\mathbf{M}^t=\mathbf{M}\right\}\)

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

\(\mathcal{Z}^{(3)}_k=\left\{\mathbf{M}\in\mathcal{P}_{k}^{d\times d}\middle|\mathbf{M}^t=\mathbf{M}\text{ and }\operatorname{div}\mathbf{M}=0\right\}\)

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 Arnold-Winther order 3 on a triangle
element = symfem.create_element("triangle", "AW", 3)

# Create Arnold-Winther order 4 on a triangle
element = symfem.create_element("triangle", "AW", 4)

Before trying this example, you must install UFL:

pip3 install UFL

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

import ufl

# Create Arnold-Winther order 3 on a triangle
element = ufl.FiniteElement("AWc", "triangle", 3)

# Create Arnold-Winther order 4 on a triangle
element = ufl.FiniteElement("AWc", "triangle", 4)