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# Degree 2 Nédélec (second kind) 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: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{11}\}$$
• 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;
$$\hat{\boldsymbol{t}}_{0}$$ is the tangent to edge 0;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{0}$$.

$$\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 3 y \left(4 x - 1\right)\\\displaystyle 9 x \left(2 x - 1\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 9 y \left(1 - 2 y\right)\\\displaystyle 3 x \left(1 - 4 y\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle \frac{3 y \left(- 6 x - y + 2\right)}{2}\\\displaystyle \frac{3 x \left(x + 6 y - 2\right)}{2}\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 3 y \left(4 x + 4 y - 3\right)\\\displaystyle 18 x^{2} + 48 x y - 27 x + 30 y^{2} - 36 y + 9\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 9 y \left(2 y - 1\right)\\\displaystyle 12 x y - 3 x + 30 y^{2} - 24 y + 3\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle \frac{3 y \left(- 6 x - 5 y + 4\right)}{2}\\\displaystyle \frac{3 x^{2}}{2} - 15 x y - 15 y^{2} + 15 y - \frac{3}{2}\end{array}\right)$$

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;
$$\hat{\boldsymbol{t}}_{2}$$ is the tangent to edge 2;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{2}$$.

$$\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 30 x^{2} + 48 x y - 36 x + 18 y^{2} - 27 y + 9\\\displaystyle 3 x \left(4 x + 4 y - 3\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 30 x^{2} + 12 x y - 24 x - 3 y + 3\\\displaystyle 9 x \left(2 x - 1\right)\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle - 15 x^{2} - 15 x y + 15 x + \frac{3 y^{2}}{2} - \frac{3}{2}\\\displaystyle \frac{3 x \left(- 5 x - 6 y + 4\right)}{2}\end{array}\right)$$

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

$$\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 12 y \left(- 4 x - y + 1\right)\\\displaystyle 12 x \left(- x - 4 y + 1\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{10}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}s_{0} - 1\\s_{1}\end{array}\right)$$
where $$R$$ is the reference element;
and $$s_{0},s_{1}$$ is a parametrisation of $$R$$.

$$\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 12 y \left(4 x + 3 y - 3\right)\\\displaystyle 12 x \left(x + 2 y - 1\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{11}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}- s_{0}\\1 - s_{1}\end{array}\right)$$
where $$R$$ is the reference element;
and $$s_{0},s_{1}$$ is a parametrisation of $$R$$.

$$\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 12 y \left(- 2 x - y + 1\right)\\\displaystyle 12 x \left(- 3 x - 4 y + 3\right)\end{array}\right)$$

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