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# Degree 3 trimmed serendipity H(curl) on a quadrilateral

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In this example:
• $$R$$ is the reference quadrilateral. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y^{3}\\\displaystyle - x y^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y^{2}\\\displaystyle - x^{2} y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2} y\\\displaystyle - x^{3}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y^{3}\\\displaystyle 3 x y^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 3 x^{2} y\\\displaystyle x^{3}\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{16}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(2 s_{0}^{2} - 3 s_{0} + 1)\hat{\boldsymbol{t}}_{0}$$
where $$e_{0}$$ is the 0th edge;
and $$\hat{\boldsymbol{t}}_{0}$$ is the tangent to edge 0.

$$\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle - 30 x^{2} y + 30 x^{2} - 9 x y^{2} + 45 x y - 36 x - 10 y^{3} + \frac{45 y^{2}}{2} - \frac{43 y}{2} + 9\\\displaystyle \frac{9 x \left(2 x y - x - 2 y + 1\right)}{2}\end{array}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(s_{0} \cdot \left(2 s_{0} - 1\right))\hat{\boldsymbol{t}}_{0}$$
where $$e_{0}$$ is the 0th edge;
and $$\hat{\boldsymbol{t}}_{0}$$ is the tangent to edge 0.

$$\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle - 30 x^{2} y + 30 x^{2} + 9 x y^{2} + 15 x y - 24 x - 10 y^{3} + \frac{27 y^{2}}{2} - \frac{13 y}{2} + 3\\\displaystyle \frac{9 x \left(- 2 x y + x + 2 y - 1\right)}{2}\end{array}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(4 s_{0} \cdot \left(1 - s_{0}\right))\hat{\boldsymbol{t}}_{0}$$
where $$e_{0}$$ is the 0th edge;
and $$\hat{\boldsymbol{t}}_{0}$$ is the tangent to edge 0.

$$\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 15 x^{2} y - 15 x^{2} - 15 x y + 15 x - 10 y^{3} + 18 y^{2} - \frac{13 y}{2} - \frac{3}{2}\\\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(2 s_{0}^{2} - 3 s_{0} + 1)\hat{\boldsymbol{t}}_{1}$$
where $$e_{1}$$ is the 1st edge;
and $$\hat{\boldsymbol{t}}_{1}$$ is the tangent to edge 1.

$$\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle \frac{9 y \left(2 x y - 2 x - y + 1\right)}{2}\\\displaystyle - 10 x^{3} - 9 x^{2} y + \frac{45 x^{2}}{2} - 30 x y^{2} + 45 x y - \frac{43 x}{2} + 30 y^{2} - 36 y + 9\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(s_{0} \cdot \left(2 s_{0} - 1\right))\hat{\boldsymbol{t}}_{1}$$
where $$e_{1}$$ is the 1st edge;
and $$\hat{\boldsymbol{t}}_{1}$$ is the tangent to edge 1.

$$\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle \frac{9 y \left(- 2 x y + 2 x + y - 1\right)}{2}\\\displaystyle - 10 x^{3} + 9 x^{2} y + \frac{27 x^{2}}{2} - 30 x y^{2} + 15 x y - \frac{13 x}{2} + 30 y^{2} - 24 y + 3\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(4 s_{0} \cdot \left(1 - s_{0}\right))\hat{\boldsymbol{t}}_{1}$$
where $$e_{1}$$ is the 1st edge;
and $$\hat{\boldsymbol{t}}_{1}$$ is the tangent to edge 1.

$$\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - 10 x^{3} + 18 x^{2} + 15 x y^{2} - 15 x y - \frac{13 x}{2} - 15 y^{2} + 15 y - \frac{3}{2}\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(2 s_{0}^{2} - 3 s_{0} + 1)\hat{\boldsymbol{t}}_{2}$$
where $$e_{2}$$ is the 2nd edge;
and $$\hat{\boldsymbol{t}}_{2}$$ is the tangent to edge 2.

$$\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle \frac{9 y \left(2 x y - 2 x - y + 1\right)}{2}\\\displaystyle \frac{x \left(20 x^{2} - 18 x y - 15 x + 60 y^{2} - 54 y + 13\right)}{2}\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(s_{0} \cdot \left(2 s_{0} - 1\right))\hat{\boldsymbol{t}}_{2}$$
where $$e_{2}$$ is the 2nd edge;
and $$\hat{\boldsymbol{t}}_{2}$$ is the tangent to edge 2.

$$\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle \frac{9 y \left(- 2 x y + 2 x + y - 1\right)}{2}\\\displaystyle \frac{x \left(20 x^{2} + 18 x y - 33 x + 60 y^{2} - 66 y + 19\right)}{2}\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(4 s_{0} \cdot \left(1 - s_{0}\right))\hat{\boldsymbol{t}}_{2}$$
where $$e_{2}$$ is the 2nd edge;
and $$\hat{\boldsymbol{t}}_{2}$$ is the tangent to edge 2.

$$\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{x \left(20 x^{2} - 24 x - 30 y^{2} + 30 y + 1\right)}{2}\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{9}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(2 s_{0}^{2} - 3 s_{0} + 1)\hat{\boldsymbol{t}}_{3}$$
where $$e_{3}$$ is the 3th edge;
and $$\hat{\boldsymbol{t}}_{3}$$ is the tangent to edge 3.

$$\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle \frac{y \left(60 x^{2} - 18 x y - 54 x + 20 y^{2} - 15 y + 13\right)}{2}\\\displaystyle \frac{9 x \left(2 x y - x - 2 y + 1\right)}{2}\end{array}\right)$$

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{10}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(s_{0} \cdot \left(2 s_{0} - 1\right))\hat{\boldsymbol{t}}_{3}$$
where $$e_{3}$$ is the 3th edge;
and $$\hat{\boldsymbol{t}}_{3}$$ is the tangent to edge 3.

$$\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle \frac{y \left(60 x^{2} + 18 x y - 66 x + 20 y^{2} - 33 y + 19\right)}{2}\\\displaystyle \frac{9 x \left(- 2 x y + x + 2 y - 1\right)}{2}\end{array}\right)$$

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{11}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(4 s_{0} \cdot \left(1 - s_{0}\right))\hat{\boldsymbol{t}}_{3}$$
where $$e_{3}$$ is the 3th edge;
and $$\hat{\boldsymbol{t}}_{3}$$ is the tangent to edge 3.

$$\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle \frac{y \left(- 30 x^{2} + 30 x + 20 y^{2} - 24 y + 1\right)}{2}\\\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{12}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right))v$$
where $$R$$ is the reference element.

$$\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 3 y \left(6 x y - 6 x + 20 y^{2} - 35 y + 15\right)\\\displaystyle 9 x \left(- 2 x y + x + 2 y - 1\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{13}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right))v$$
where $$R$$ is the reference element.

$$\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 9 y \left(- 2 x y + 2 x + y - 1\right)\\\displaystyle 3 x \left(20 x^{2} + 6 x y - 35 x - 6 y + 15\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{14}:\mathbf{v}\mapsto\displaystyle\int_{R}(\left(\begin{array}{c}\displaystyle 0\\\displaystyle - 2 s_{0}\end{array}\right))v$$
where $$R$$ is the reference element.

$$\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 30 x \left(2 x^{2} - 3 x + 1\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{15}:\mathbf{v}\mapsto\displaystyle\int_{R}(\left(\begin{array}{c}\displaystyle s_{0}\\\displaystyle - t_{1}\end{array}\right))v$$
where $$R$$ is the reference element.

$$\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 18 y \left(- 2 x y + 2 x + y - 1\right)\\\displaystyle 18 x \left(2 x y - x - 2 y + 1\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{16}:\mathbf{v}\mapsto\displaystyle\int_{R}(\left(\begin{array}{c}\displaystyle 2 s_{1}\\\displaystyle 0\end{array}\right))v$$
where $$R$$ is the reference element.

$$\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle 30 y \left(- 2 y^{2} + 3 y - 1\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 0 of the reference element.