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# Degree 2 nonconforming Arnold–Winther on a triangle

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In this example:
• $$R$$ is the reference triangle. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$\left(\begin{array}{cc}\displaystyle 1&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle 1\\\displaystyle 1&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle x&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle x\\\displaystyle x&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle y&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle y\\\displaystyle y&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle y^{2}\\\displaystyle y^{2}&\displaystyle - 2 y^{2}\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle - 2 x^{2}&\displaystyle x^{2}\\\displaystyle x^{2}&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle - 2 x y&\displaystyle x y\\\displaystyle x y&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle x \left(x - y\right)&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle x^{2}&\displaystyle 0\\\displaystyle 0&\displaystyle x y\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle x^{2}&\displaystyle 0\\\displaystyle 0&\displaystyle y^{2}\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{14}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{0}}(1 - s_{0})\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle - \frac{\sqrt{2}}{2}\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle - \frac{\sqrt{2}}{2}\end{array}\right)$$
where $$e_{0}$$ is the 0th edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{0}$$.

$$\displaystyle \mathbf{\Phi}_{0} = \left(\begin{array}{cc}\displaystyle \frac{3 x \left(3 x - 2 y - 1\right)}{2}&\displaystyle \frac{3 x^{2}}{2} + 3 x y - \frac{3 x}{2} + \frac{3 y^{2}}{2} - \frac{3 y}{2} + \frac{1}{4}\\\displaystyle \frac{3 x^{2}}{2} + 3 x y - \frac{3 x}{2} + \frac{3 y^{2}}{2} - \frac{3 y}{2} + \frac{1}{4}&\displaystyle \frac{3 y \left(6 x - y - 1\right)}{2}\end{array}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{1}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{0}}(1 - s_{0})\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle \frac{\sqrt{2}}{2}\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle - \frac{\sqrt{2}}{2}\end{array}\right)$$
where $$e_{0}$$ is the 0th edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{0}$$.

$$\displaystyle \mathbf{\Phi}_{1} = \left(\begin{array}{cc}\displaystyle \frac{3 x \left(7 x + 6 y - 5\right)}{2}&\displaystyle - \frac{9 x^{2}}{2} - 9 x y + \frac{9 x}{2} - \frac{9 y^{2}}{2} + \frac{9 y}{2} - \frac{3}{4}\\\displaystyle - \frac{9 x^{2}}{2} - 9 x y + \frac{9 x}{2} - \frac{9 y^{2}}{2} + \frac{9 y}{2} - \frac{3}{4}&\displaystyle \frac{3 y \left(- 2 x + 3 y - 1\right)}{2}\end{array}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{2}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{0}}(s_{0})\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle - \frac{\sqrt{2}}{2}\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle - \frac{\sqrt{2}}{2}\end{array}\right)$$
where $$e_{0}$$ is the 0th edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{0}$$.

$$\displaystyle \mathbf{\Phi}_{2} = \left(\begin{array}{cc}\displaystyle \frac{3 x \left(x + 10 y - 3\right)}{2}&\displaystyle - \frac{3 x^{2}}{2} - 3 x y + \frac{3 x}{2} - \frac{3 y^{2}}{2} + \frac{3 y}{2} - \frac{1}{4}\\\displaystyle - \frac{3 x^{2}}{2} - 3 x y + \frac{3 x}{2} - \frac{3 y^{2}}{2} + \frac{3 y}{2} - \frac{1}{4}&\displaystyle \frac{3 y \left(2 x + 5 y - 3\right)}{2}\end{array}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{3}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{0}}(s_{0})\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle \frac{\sqrt{2}}{2}\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle - \frac{\sqrt{2}}{2}\end{array}\right)$$
where $$e_{0}$$ is the 0th edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{0}$$.

$$\displaystyle \mathbf{\Phi}_{3} = \left(\begin{array}{cc}\displaystyle \frac{3 x \left(3 x + 14 y - 5\right)}{2}&\displaystyle - \frac{9 x^{2}}{2} - 9 x y + \frac{9 x}{2} - \frac{9 y^{2}}{2} + \frac{9 y}{2} - \frac{3}{4}\\\displaystyle - \frac{9 x^{2}}{2} - 9 x y + \frac{9 x}{2} - \frac{9 y^{2}}{2} + \frac{9 y}{2} - \frac{3}{4}&\displaystyle \frac{3 y \left(6 x - y - 1\right)}{2}\end{array}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{4}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{1}}(1 - s_{0})\left(\begin{array}{c}\displaystyle -1\\\displaystyle 0\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle -1\\\displaystyle 0\end{array}\right)$$
where $$e_{1}$$ is the 1st edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{1}$$.

$$\displaystyle \mathbf{\Phi}_{4} = \left(\begin{array}{cc}\displaystyle 21 x^{2} + 42 x y - 27 x - 6 y + 4&\displaystyle - 9 x^{2} - 18 x y + 9 x - 9 y^{2} + 9 y - \frac{3}{2}\\\displaystyle - 9 x^{2} - 18 x y + 9 x - 9 y^{2} + 9 y - \frac{3}{2}&\displaystyle 9 y \left(2 x + y - 1\right)\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{5}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{1}}(1 - s_{0})\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle -1\\\displaystyle 0\end{array}\right)$$
where $$e_{1}$$ is the 1st edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{1}$$.

$$\displaystyle \mathbf{\Phi}_{5} = \left(\begin{array}{cc}\displaystyle 3 x \left(- 5 x - 18 y + 7\right)&\displaystyle 3 x^{2} + 30 x y - 3 x + 27 y^{2} - 21 y + \frac{1}{2}\\\displaystyle 3 x^{2} + 30 x y - 3 x + 27 y^{2} - 21 y + \frac{1}{2}&\displaystyle 3 y \left(- 10 x - 9 y + 7\right)\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{6}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{1}}(s_{0})\left(\begin{array}{c}\displaystyle -1\\\displaystyle 0\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle -1\\\displaystyle 0\end{array}\right)$$
where $$e_{1}$$ is the 1st edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{1}$$.

$$\displaystyle \mathbf{\Phi}_{6} = \left(\begin{array}{cc}\displaystyle - 6 x^{2} - 36 x y + 12 x + 6 y - 2&\displaystyle 6 x^{2} + 12 x y - 6 x + 6 y^{2} - 6 y + 1\\\displaystyle 6 x^{2} + 12 x y - 6 x + 6 y^{2} - 6 y + 1&\displaystyle 6 y \left(- 2 x - y + 1\right)\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{7}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{1}}(s_{0})\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle -1\\\displaystyle 0\end{array}\right)$$
where $$e_{1}$$ is the 1st edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{1}$$.

$$\displaystyle \mathbf{\Phi}_{7} = \left(\begin{array}{cc}\displaystyle 3 x \left(3 x + 14 y - 5\right)&\displaystyle - 9 x^{2} - 18 x y + 9 x - 21 y^{2} + 15 y - \frac{3}{2}\\\displaystyle - 9 x^{2} - 18 x y + 9 x - 21 y^{2} + 15 y - \frac{3}{2}&\displaystyle 3 y \left(6 x + 7 y - 5\right)\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{8}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{2}}(1 - s_{0})\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$
where $$e_{2}$$ is the 2nd edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{2}$$.

$$\displaystyle \mathbf{\Phi}_{8} = \left(\begin{array}{cc}\displaystyle 9 x \left(- x - 2 y + 1\right)&\displaystyle 9 x^{2} + 18 x y - 9 x + 9 y^{2} - 9 y + \frac{3}{2}\\\displaystyle 9 x^{2} + 18 x y - 9 x + 9 y^{2} - 9 y + \frac{3}{2}&\displaystyle 6 x y - 6 x + 3 y^{2} - 9 y + 4\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{9}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{2}}(1 - s_{0})\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$
where $$e_{2}$$ is the 2nd edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{2}$$.

$$\displaystyle \mathbf{\Phi}_{9} = \left(\begin{array}{cc}\displaystyle 3 x \left(- x - 10 y + 3\right)&\displaystyle 3 x^{2} + 30 x y - 9 x + 27 y^{2} - 27 y + \frac{9}{2}\\\displaystyle 3 x^{2} + 30 x y - 9 x + 27 y^{2} - 27 y + \frac{9}{2}&\displaystyle 3 y \left(- 2 x - 5 y + 3\right)\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{10}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{2}}(s_{0})\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$
where $$e_{2}$$ is the 2nd edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{2}$$.

$$\displaystyle \mathbf{\Phi}_{10} = \left(\begin{array}{cc}\displaystyle 6 x \left(x + 2 y - 1\right)&\displaystyle - 6 x^{2} - 12 x y + 6 x - 6 y^{2} + 6 y - 1\\\displaystyle - 6 x^{2} - 12 x y + 6 x - 6 y^{2} + 6 y - 1&\displaystyle - 12 x y + 6 x + 6 y^{2} - 2\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{11}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{2}}(s_{0})\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$
where $$e_{2}$$ is the 2nd edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{2}$$.

$$\displaystyle \mathbf{\Phi}_{11} = \left(\begin{array}{cc}\displaystyle 3 x \left(- x + 6 y - 1\right)&\displaystyle 3 x^{2} - 18 x y + 3 x - 9 y^{2} + 9 y - \frac{3}{2}\\\displaystyle 3 x^{2} - 18 x y + 3 x - 9 y^{2} + 9 y - \frac{3}{2}&\displaystyle 3 y \left(- 2 x + 3 y - 1\right)\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{12}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{cc}\displaystyle 1&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right))v$$
where $$R$$ is the reference element.

$$\displaystyle \mathbf{\Phi}_{12} = \left(\begin{array}{cc}\displaystyle 12 x \left(- 3 x - 4 y + 3\right)&\displaystyle 12 x^{2} + 24 x y - 12 x + 12 y^{2} - 12 y + 2\\\displaystyle 12 x^{2} + 24 x y - 12 x + 12 y^{2} - 12 y + 2&\displaystyle 12 y \left(- 2 x - y + 1\right)\end{array}\right)$$

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

$$\displaystyle \mathbf{\Phi}_{13} = \left(\begin{array}{cc}\displaystyle 12 x \left(- x - 2 y + 1\right)&\displaystyle - 12 x^{2} + 12 x - 12 y^{2} + 12 y - 2\\\displaystyle - 12 x^{2} + 12 x - 12 y^{2} + 12 y - 2&\displaystyle 12 y \left(- 2 x - y + 1\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{14}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 1\end{array}\right))v$$
where $$R$$ is the reference element.

$$\displaystyle \mathbf{\Phi}_{14} = \left(\begin{array}{cc}\displaystyle 12 x \left(x + 2 y - 1\right)&\displaystyle - 12 x^{2} - 24 x y + 12 x - 12 y^{2} + 12 y - 2\\\displaystyle - 12 x^{2} - 24 x y + 12 x - 12 y^{2} + 12 y - 2&\displaystyle 12 y \left(1 - y\right)\end{array}\right)$$

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