Tiniest tensor H(curl)

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

\(\mathcal{Z}^{(35)}_k=\left\{\left(\begin{array}{c}\tilde{P}_k(x)\tilde{B}_{k+1}(y)\\-\tilde{B}_{k+1}(x)\tilde{P}_{k}(y)\end{array}\right),\left(\begin{array}{c}\tilde{P}_k(0)\tilde{B}_{k+1}(y)\\-\tilde{B}_{k+1}(0)\tilde{P}_{k}(y)\end{array}\right),\left(\begin{array}{c}\tilde{P}_k(x)\tilde{B}_{k+1}(0)\\-\tilde{B}_{k+1}(x)\tilde{P}_{k}(0)\end{array}\right)\right\}] \)
where

• \(\tilde{P}_k\) is the degree \(k\) Legendre polynomial on \([0,1]\)
• \(\tilde{B}_k\) is \((\tilde{P}_{k+2}-\tilde{P}_k)/(4k+6)\)

\(\mathcal{Z}^{(36)}_k=\left\{f\boldsymbol{g}\middle|f\in\{x,1-x\},\boldsymbol{g}\in\left\{\left(\begin{array}{c}\tilde{P}_k(y)\tilde{B}_{k+1}(z)\\-\tilde{B}_{k+1}(y)\tilde{P}_{k}(z)\end{array}\right),\left(\begin{array}{c}\tilde{P}_k(0)\tilde{B}_{k+1}(z)\\-\tilde{B}_{k+1}(0)\tilde{P}_{k}(z)\end{array}\right),\left(\begin{array}{c}\tilde{P}_k(y)\tilde{B}_{k+1}(0)\\-\tilde{B}_{k+1}(y)\tilde{P}_{k}(0)\end{array}\right),\right\}\right\} \)
where

• \(\tilde{P}_k\) is the degree \(k\) Legendre polynomial on \([0,1]\)
• \(\tilde{B}_k\) is \((\tilde{P}_{k+2}-\tilde{P}_k)/(4k+6)\)

\(\mathcal{Z}^{(37)}_k=\left\{f\boldsymbol{g}\middle|f\in\{y,1-y\},\boldsymbol{g}\in\left\{\left(\begin{array}{c}\tilde{P}_k(x)\tilde{B}_{k+1}(z)\\-\tilde{B}_{k+1}(x)\tilde{P}_{k}(z)\end{array}\right),\left(\begin{array}{c}\tilde{P}_k(0)\tilde{B}_{k+1}(z)\\-\tilde{B}_{k+1}(0)\tilde{P}_{k}(z)\end{array}\right),\left(\begin{array}{c}\tilde{P}_k(x)\tilde{B}_{k+1}(0)\\-\tilde{B}_{k+1}(x)\tilde{P}_{k}(0)\end{array}\right),\right\}\right\}\)

\(\mathcal{Z}^{(38)}_k=\left\{f\boldsymbol{g}\middle|f\in\{z,1-z\},\boldsymbol{g}\in\left\{\left(\begin{array}{c}\tilde{P}_k(x)\tilde{B}_{k+1}(y)\\-\tilde{B}_{k+1}(x)\tilde{P}_{k}(y)\end{array}\right),\left(\begin{array}{c}\tilde{P}_k(0)\tilde{B}_{k+1}(y)\\-\tilde{B}_{k+1}(0)\tilde{P}_{k}(y)\end{array}\right),\left(\begin{array}{c}\tilde{P}_k(x)\tilde{B}_{k+1}(0)\\-\tilde{B}_{k+1}(x)\tilde{P}_{k}(0)\end{array}\right),\right\}\right\}\)

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 Tiniest tensor H(curl) order 1 on a quadrilateral
element = symfem.create_element("quadrilateral", "TNTcurl", 1)

# Create Tiniest tensor H(curl) order 2 on a quadrilateral
element = symfem.create_element("quadrilateral", "TNTcurl", 2)

# Create Tiniest tensor H(curl) order 3 on a quadrilateral
element = symfem.create_element("quadrilateral", "TNTcurl", 3)

# Create Tiniest tensor H(curl) order 1 on a hexahedron
element = symfem.create_element("hexahedron", "TNTcurl", 1)