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# Degree 2 Regge 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: $$\left(\begin{array}{cc}\displaystyle 1&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle 1\\\displaystyle 1&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle x&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle x\\\displaystyle x&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle x^{2}&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle x^{2}\\\displaystyle x^{2}&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle x^{2}\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle y&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle y\\\displaystyle y&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle x y&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle x y\\\displaystyle x y&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle x y\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle y^{2}&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle y^{2}\\\displaystyle y^{2}&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle y^{2}\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{17}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\mathbf{V}\mapsto\left(\begin{array}{c}-1\\1\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{3}{4},\tfrac{1}{4})\left(\begin{array}{c}-1\\1\end{array}\right)$$

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

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{1}:\mathbf{V}\mapsto\left(\begin{array}{c}-1\\1\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{2},\tfrac{1}{2})\left(\begin{array}{c}-1\\1\end{array}\right)$$

$$\displaystyle \mathbf{\Phi}_{1} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle - 8 x y + 2 x + 2 y - \frac{1}{2}\\\displaystyle - 8 x y + 2 x + 2 y - \frac{1}{2}&\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{2}:\mathbf{V}\mapsto\left(\begin{array}{c}-1\\1\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{4},\tfrac{3}{4})\left(\begin{array}{c}-1\\1\end{array}\right)$$

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

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{3}:\mathbf{V}\mapsto\left(\begin{array}{c}0\\1\end{array}\right)^{\text{t}}\mathbf{V}(0,\tfrac{1}{4})\left(\begin{array}{c}0\\1\end{array}\right)$$

$$\displaystyle \mathbf{\Phi}_{3} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle 4 x^{2} + 8 x y - 5 x + 4 y^{2} - 5 y + \frac{3}{2}\\\displaystyle 4 x^{2} + 8 x y - 5 x + 4 y^{2} - 5 y + \frac{3}{2}&\displaystyle 8 x^{2} + 16 x y - 10 x + 8 y^{2} - 10 y + 3\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{4}:\mathbf{V}\mapsto\left(\begin{array}{c}0\\1\end{array}\right)^{\text{t}}\mathbf{V}(0,\tfrac{1}{2})\left(\begin{array}{c}0\\1\end{array}\right)$$

$$\displaystyle \mathbf{\Phi}_{4} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle - 8 x y + 2 x - 8 y^{2} + 8 y - \frac{3}{2}\\\displaystyle - 8 x y + 2 x - 8 y^{2} + 8 y - \frac{3}{2}&\displaystyle - 16 x y + 4 x - 16 y^{2} + 16 y - 3\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{5}:\mathbf{V}\mapsto\left(\begin{array}{c}0\\1\end{array}\right)^{\text{t}}\mathbf{V}(0,\tfrac{3}{4})\left(\begin{array}{c}0\\1\end{array}\right)$$

$$\displaystyle \mathbf{\Phi}_{5} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle 4 y^{2} - 3 y + \frac{1}{2}\\\displaystyle 4 y^{2} - 3 y + \frac{1}{2}&\displaystyle 8 y^{2} - 6 y + 1\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{6}:\mathbf{V}\mapsto\left(\begin{array}{c}1\\0\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{4},0)\left(\begin{array}{c}1\\0\end{array}\right)$$

$$\displaystyle \mathbf{\Phi}_{6} = \left(\begin{array}{cc}\displaystyle 8 x^{2} + 16 x y - 10 x + 8 y^{2} - 10 y + 3&\displaystyle 4 x^{2} + 8 x y - 5 x + 4 y^{2} - 5 y + \frac{3}{2}\\\displaystyle 4 x^{2} + 8 x y - 5 x + 4 y^{2} - 5 y + \frac{3}{2}&\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{7}:\mathbf{V}\mapsto\left(\begin{array}{c}1\\0\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{2},0)\left(\begin{array}{c}1\\0\end{array}\right)$$

$$\displaystyle \mathbf{\Phi}_{7} = \left(\begin{array}{cc}\displaystyle - 16 x^{2} - 16 x y + 16 x + 4 y - 3&\displaystyle - 8 x^{2} - 8 x y + 8 x + 2 y - \frac{3}{2}\\\displaystyle - 8 x^{2} - 8 x y + 8 x + 2 y - \frac{3}{2}&\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{8}:\mathbf{V}\mapsto\left(\begin{array}{c}1\\0\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{3}{4},0)\left(\begin{array}{c}1\\0\end{array}\right)$$

$$\displaystyle \mathbf{\Phi}_{8} = \left(\begin{array}{cc}\displaystyle 8 x^{2} - 6 x + 1&\displaystyle 4 x^{2} - 3 x + \frac{1}{2}\\\displaystyle 4 x^{2} - 3 x + \frac{1}{2}&\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{9}:\mathbf{V}\mapsto\left(\begin{array}{c}1\\0\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{4},\tfrac{1}{4})\left(\begin{array}{c}1\\0\end{array}\right)$$

$$\displaystyle \mathbf{\Phi}_{9} = \left(\begin{array}{cc}\displaystyle 4 y \left(- 4 x - 4 y + 3\right)&\displaystyle 2 y \left(- 4 x - 4 y + 3\right)\\\displaystyle 2 y \left(- 4 x - 4 y + 3\right)&\displaystyle 0\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{10}:\mathbf{V}\mapsto\left(\begin{array}{c}0\\1\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{4},\tfrac{1}{4})\left(\begin{array}{c}0\\1\end{array}\right)$$

$$\displaystyle \mathbf{\Phi}_{10} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle 2 x \left(- 4 x - 4 y + 3\right)\\\displaystyle 2 x \left(- 4 x - 4 y + 3\right)&\displaystyle 4 x \left(- 4 x - 4 y + 3\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{11}:\mathbf{V}\mapsto\left(\begin{array}{c}-1\\1\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{4},\tfrac{1}{4})\left(\begin{array}{c}-1\\1\end{array}\right)$$

$$\displaystyle \mathbf{\Phi}_{11} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle - 4 x^{2} - 8 x y + 7 x - 4 y^{2} + 7 y - 3\\\displaystyle - 4 x^{2} - 8 x y + 7 x - 4 y^{2} + 7 y - 3&\displaystyle 0\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{12}:\mathbf{V}\mapsto\left(\begin{array}{c}1\\0\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{2},\tfrac{1}{4})\left(\begin{array}{c}1\\0\end{array}\right)$$

$$\displaystyle \mathbf{\Phi}_{12} = \left(\begin{array}{cc}\displaystyle 4 y \left(4 x - 1\right)&\displaystyle 2 y \left(4 x - 1\right)\\\displaystyle 2 y \left(4 x - 1\right)&\displaystyle 0\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{13}:\mathbf{V}\mapsto\left(\begin{array}{c}0\\1\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{2},\tfrac{1}{4})\left(\begin{array}{c}0\\1\end{array}\right)$$

$$\displaystyle \mathbf{\Phi}_{13} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle x \left(4 x - 1\right)\\\displaystyle x \left(4 x - 1\right)&\displaystyle 2 x \left(4 x - 1\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{14}:\mathbf{V}\mapsto\left(\begin{array}{c}-1\\1\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{2},\tfrac{1}{4})\left(\begin{array}{c}-1\\1\end{array}\right)$$

$$\displaystyle \mathbf{\Phi}_{14} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle 8 x^{2} + 8 x y - 10 x - 2 y + 2\\\displaystyle 8 x^{2} + 8 x y - 10 x - 2 y + 2&\displaystyle 0\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{15}:\mathbf{V}\mapsto\left(\begin{array}{c}1\\0\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{4},\tfrac{1}{2})\left(\begin{array}{c}1\\0\end{array}\right)$$

$$\displaystyle \mathbf{\Phi}_{15} = \left(\begin{array}{cc}\displaystyle 2 y \left(4 y - 1\right)&\displaystyle y \left(4 y - 1\right)\\\displaystyle y \left(4 y - 1\right)&\displaystyle 0\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{16}:\mathbf{V}\mapsto\left(\begin{array}{c}0\\1\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{4},\tfrac{1}{2})\left(\begin{array}{c}0\\1\end{array}\right)$$

$$\displaystyle \mathbf{\Phi}_{16} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle 2 x \left(4 y - 1\right)\\\displaystyle 2 x \left(4 y - 1\right)&\displaystyle 4 x \left(4 y - 1\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{17}:\mathbf{V}\mapsto\left(\begin{array}{c}-1\\1\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{4},\tfrac{1}{2})\left(\begin{array}{c}-1\\1\end{array}\right)$$

$$\displaystyle \mathbf{\Phi}_{17} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle 8 x y - 2 x + 8 y^{2} - 10 y + 2\\\displaystyle 8 x y - 2 x + 8 y^{2} - 10 y + 2&\displaystyle 0\end{array}\right)$$

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