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# Degree 3 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: $$\left(\begin{array}{cc}\displaystyle 1&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle 1\\\displaystyle 1&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle x&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle x\\\displaystyle x&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle x^{2}&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle x^{2}\\\displaystyle x^{2}&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle x^{2}\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle y&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle y\\\displaystyle y&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle x y&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle x y\\\displaystyle x y&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle x y\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle y^{2}&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle y^{2}\\\displaystyle y^{2}&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle y^{2}\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 20 y^{3}&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 12 x y^{2}&\displaystyle - 4 y^{3}\\\displaystyle - 4 y^{3}&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 6 x^{2} y&\displaystyle - 6 x y^{2}\\\displaystyle - 6 x y^{2}&\displaystyle - 2 y^{3}\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 2 x^{3}&\displaystyle - 6 x^{2} y\\\displaystyle - 6 x^{2} y&\displaystyle - 6 x y^{2}\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle x^{3}\\\displaystyle x^{3}&\displaystyle - 3 x^{2} y\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle x^{3}\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{23}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\mathbf{V}\mapsto\left(\begin{array}{c}1\\0\end{array}\right)^{\text{t}}\mathbf{V}(0,0)\left(\begin{array}{c}1\\0\end{array}\right)$$

$$\displaystyle \mathbf{\Phi}_{0} = \left(\begin{array}{cc}\displaystyle - \frac{5 x^{3}}{2} + \frac{3 x^{2}}{2} - 10 y^{3} + 18 y^{2} - 9 y + 1&\displaystyle \frac{3 x y \left(5 x - 2\right)}{2}\\\displaystyle \frac{3 x y \left(5 x - 2\right)}{2}&\displaystyle \frac{3 y \left(5 x y - 4 x - y + 1\right)}{2}\end{array}\right)$$

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

$$\displaystyle \mathbf{\Phi}_{1} = \left(\begin{array}{cc}\displaystyle 3 x \left(- 10 x^{2} - 15 x y + 7 x + 10 y^{2} - 12 y + 3\right)&\displaystyle - 10 x^{3} + 90 x^{2} y + 18 x^{2} + 45 x y^{2} - 42 x y - 9 x - 10 y^{3} + 18 y^{2} - 9 y + 1\\\displaystyle - 10 x^{3} + 90 x^{2} y + 18 x^{2} + 45 x y^{2} - 42 x y - 9 x - 10 y^{3} + 18 y^{2} - 9 y + 1&\displaystyle 3 y \left(10 x^{2} + 30 x y - 36 x + 5 y^{2} - 17 y + 12\right)\end{array}\right)$$

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

$$\displaystyle \mathbf{\Phi}_{2} = \left(\begin{array}{cc}\displaystyle \frac{15 x^{2} \left(- x - 2 y + 1\right)}{2}&\displaystyle \frac{15 x y \left(3 x + 2 y - 2\right)}{2}\\\displaystyle \frac{15 x y \left(3 x + 2 y - 2\right)}{2}&\displaystyle - 10 x^{3} + 18 x^{2} + \frac{45 x y^{2}}{2} - 18 x y - 9 x + 5 y^{3} - \frac{27 y^{2}}{2} + \frac{15 y}{2} + 1\end{array}\right)$$

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

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

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

$$\displaystyle \mathbf{\Phi}_{4} = \left(\begin{array}{cc}\displaystyle \frac{45 x^{2} \left(x + 2 y - 1\right)}{2}&\displaystyle \frac{x \left(20 x^{2} - 135 x y - 24 x - 90 y^{2} + 90 y + 6\right)}{2}\\\displaystyle \frac{x \left(20 x^{2} - 135 x y - 24 x - 90 y^{2} + 90 y + 6\right)}{2}&\displaystyle \frac{3 y \left(- 20 x^{2} - 45 x y + 52 x - 10 y^{2} + 27 y - 17\right)}{2}\end{array}\right)$$

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

$$\displaystyle \mathbf{\Phi}_{5} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle x \left(10 x^{2} - 12 x + 3\right)\end{array}\right)$$

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

$$\displaystyle \mathbf{\Phi}_{6} = \left(\begin{array}{cc}\displaystyle y \left(10 y^{2} - 12 y + 3\right)&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)$$

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

$$\displaystyle \mathbf{\Phi}_{7} = \left(\begin{array}{cc}\displaystyle \frac{3 x \left(5 x^{2} - 3 x - 20 y^{2} + 16 y - 2\right)}{2}&\displaystyle \frac{y \left(- 45 x^{2} + 18 x + 20 y^{2} - 24 y + 6\right)}{2}\\\displaystyle \frac{y \left(- 45 x^{2} + 18 x + 20 y^{2} - 24 y + 6\right)}{2}&\displaystyle \frac{9 y \left(- 5 x y + 4 x + y - 1\right)}{2}\end{array}\right)$$

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

$$\displaystyle \mathbf{\Phi}_{8} = \left(\begin{array}{cc}\displaystyle \frac{15 x^{2} \left(x + 2 y - 1\right)}{2}&\displaystyle \frac{15 x y \left(- 3 x - 2 y + 2\right)}{2}\\\displaystyle \frac{15 x y \left(- 3 x - 2 y + 2\right)}{2}&\displaystyle \frac{y \left(- 45 x y + 36 x - 10 y^{2} + 27 y - 15\right)}{2}\end{array}\right)$$

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

$$\displaystyle \mathbf{\Phi}_{9} = \left(\begin{array}{cc}\displaystyle 3 x \left(5 x^{2} + 15 x y - 4 x - 1\right)&\displaystyle 9 x y \left(- 5 x - 5 y + 4\right)\\\displaystyle 9 x y \left(- 5 x - 5 y + 4\right)&\displaystyle 3 y \left(- 15 x y + 16 x - 5 y^{2} + 12 y - 7\right)\end{array}\right)$$

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

$$\displaystyle \mathbf{\Phi}_{10} = \left(\begin{array}{cc}\displaystyle 3 x \left(- 10 x^{2} - 15 x y + 11 x - 1\right)&\displaystyle 9 x y \left(10 x + 5 y - 6\right)\\\displaystyle 9 x y \left(10 x + 5 y - 6\right)&\displaystyle 3 y \left(30 x y - 28 x + 5 y^{2} - 15 y + 10\right)\end{array}\right)$$

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

$$\displaystyle \mathbf{\Phi}_{11} = \left(\begin{array}{cc}\displaystyle 3 x \left(- 5 x^{2} - 15 x y + 6 x + 4 y - 1\right)&\displaystyle 9 x y \left(5 x + 5 y - 4\right)\\\displaystyle 9 x y \left(5 x + 5 y - 4\right)&\displaystyle 3 y \left(15 x y - 12 x + 5 y^{2} - 10 y + 5\right)\end{array}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{12}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{0}}(s_{0})\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle \frac{\sqrt{2}}{2}\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle - \frac{\sqrt{2}}{2}\end{array}\right)$$
where $$e_{0}$$ is the 0th edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{0}$$.

$$\displaystyle \mathbf{\Phi}_{12} = \left(\begin{array}{cc}\displaystyle 3 x \left(10 x^{2} + 15 x y - 9 x + 4 y - 1\right)&\displaystyle 9 x y \left(- 10 x - 5 y + 6\right)\\\displaystyle 9 x y \left(- 10 x - 5 y + 6\right)&\displaystyle 3 y \left(- 30 x y + 24 x - 5 y^{2} + 13 y - 8\right)\end{array}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{13}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{1}}(1 - s_{0})\left(\begin{array}{c}\displaystyle -1\\\displaystyle 0\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle -1\\\displaystyle 0\end{array}\right)$$
where $$e_{1}$$ is the 1st edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{1}$$.

$$\displaystyle \mathbf{\Phi}_{13} = \left(\begin{array}{cc}\displaystyle 15 x^{3} + 3 x^{2} + 24 x y - 18 x + 60 y^{3} - 96 y^{2} + 36 y&\displaystyle 9 x y \left(2 - 5 x\right)\\\displaystyle 9 x y \left(2 - 5 x\right)&\displaystyle 9 y \left(- 5 x y + 4 x + y - 1\right)\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{14}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{1}}(1 - s_{0})\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle -1\\\displaystyle 0\end{array}\right)$$
where $$e_{1}$$ is the 1st edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{1}$$.

$$\displaystyle \mathbf{\Phi}_{14} = \left(\begin{array}{cc}\displaystyle 3 x \left(- 5 x^{2} + 30 x y - 7 x + 60 y^{2} - 64 y + 12\right)&\displaystyle 3 y \left(15 x^{2} - 30 x y + 14 x - 20 y^{2} + 32 y - 12\right)\\\displaystyle 3 y \left(15 x^{2} - 30 x y + 14 x - 20 y^{2} + 32 y - 12\right)&\displaystyle 3 y \left(15 x y - 20 x - 10 y^{2} + 7 y + 3\right)\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{15}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{1}}(s_{0})\left(\begin{array}{c}\displaystyle -1\\\displaystyle 0\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle -1\\\displaystyle 0\end{array}\right)$$
where $$e_{1}$$ is the 1st edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{1}$$.

$$\displaystyle \mathbf{\Phi}_{15} = \left(\begin{array}{cc}\displaystyle - 15 x^{3} + 9 x^{2} - 24 x y + 6 x - 60 y^{3} + 84 y^{2} - 24 y&\displaystyle 9 x y \left(5 x - 2\right)\\\displaystyle 9 x y \left(5 x - 2\right)&\displaystyle 9 y \left(5 x y - 4 x - y + 1\right)\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{16}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{1}}(s_{0})\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle -1\\\displaystyle 0\end{array}\right)$$
where $$e_{1}$$ is the 1st edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{1}$$.

$$\displaystyle \mathbf{\Phi}_{16} = \left(\begin{array}{cc}\displaystyle 3 x \left(5 x^{2} - 30 x y + 3 x - 60 y^{2} + 56 y - 8\right)&\displaystyle 3 y \left(- 15 x^{2} + 30 x y - 6 x + 20 y^{2} - 28 y + 8\right)\\\displaystyle 3 y \left(- 15 x^{2} + 30 x y - 6 x + 20 y^{2} - 28 y + 8\right)&\displaystyle 3 y \left(- 15 x y + 12 x + 10 y^{2} - 11 y + 1\right)\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{17}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{2}}(1 - s_{0})\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$
where $$e_{2}$$ is the 2nd edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{2}$$.

$$\displaystyle \mathbf{\Phi}_{17} = \left(\begin{array}{cc}\displaystyle 45 x^{2} \left(x + 2 y - 1\right)&\displaystyle 45 x y \left(- 3 x - 2 y + 2\right)\\\displaystyle 45 x y \left(- 3 x - 2 y + 2\right)&\displaystyle 60 x^{3} - 96 x^{2} - 135 x y^{2} + 132 x y + 36 x - 30 y^{3} + 93 y^{2} - 63 y\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{18}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{2}}(1 - s_{0})\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$
where $$e_{2}$$ is the 2nd edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{2}$$.

$$\displaystyle \mathbf{\Phi}_{18} = \left(\begin{array}{cc}\displaystyle 3 x \left(55 x^{2} + 120 x y - 49 x + 8 y - 6\right)&\displaystyle 3 x \left(20 x^{2} - 165 x y - 32 x - 120 y^{2} + 106 y + 12\right)\\\displaystyle 3 x \left(20 x^{2} - 165 x y - 32 x - 120 y^{2} + 106 y + 12\right)&\displaystyle 3 y \left(- 60 x^{2} - 165 x y + 196 x - 40 y^{2} + 109 y - 69\right)\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{19}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{2}}(s_{0})\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$
where $$e_{2}$$ is the 2nd edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{2}$$.

$$\displaystyle \mathbf{\Phi}_{19} = \left(\begin{array}{cc}\displaystyle 45 x^{2} \left(- x - 2 y + 1\right)&\displaystyle 45 x y \left(3 x + 2 y - 2\right)\\\displaystyle 45 x y \left(3 x + 2 y - 2\right)&\displaystyle - 60 x^{3} + 84 x^{2} + 135 x y^{2} - 132 x y - 24 x + 30 y^{3} - 81 y^{2} + 51 y\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{20}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{2}}(s_{0})\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$
where $$e_{2}$$ is the 2nd edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{2}$$.

$$\displaystyle \mathbf{\Phi}_{20} = \left(\begin{array}{cc}\displaystyle 3 x \left(- 55 x^{2} - 120 x y + 53 x + 2\right)&\displaystyle 3 x \left(- 20 x^{2} + 165 x y + 28 x + 120 y^{2} - 114 y - 8\right)\\\displaystyle 3 x \left(- 20 x^{2} + 165 x y + 28 x + 120 y^{2} - 114 y - 8\right)&\displaystyle 3 y \left(60 x^{2} + 165 x y - 188 x + 40 y^{2} - 105 y + 65\right)\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{21}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{cc}\displaystyle 1&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right))v$$
where $$R$$ is the reference element.

$$\displaystyle \mathbf{\Phi}_{21} = \left(\begin{array}{cc}\displaystyle 24 x \left(- x - y + 1\right)&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{22}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 1\\\displaystyle 0&\displaystyle 0\end{array}\right))v$$
where $$R$$ is the reference element.

$$\displaystyle \mathbf{\Phi}_{22} = \left(\begin{array}{cc}\displaystyle 24 x \left(- x - 2 y + 1\right)&\displaystyle 24 x y\\\displaystyle 24 x y&\displaystyle 24 y \left(- 2 x - y + 1\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{23}:v\mapsto\displaystyle\int_{R}(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 1\end{array}\right))v$$
where $$R$$ is the reference element.

$$\displaystyle \mathbf{\Phi}_{23} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 24 y \left(- x - y + 1\right)\end{array}\right)$$

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