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# Degree 2 Lagrange 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: $$1$$, $$y$$, $$y^{2}$$, $$x$$, $$x y$$, $$x y^{2}$$, $$x^{2}$$, $$x^{2} y$$, $$x^{2} y^{2}$$
• $$\mathcal{L}=\{l_0,...,l_{8}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto v(0,0)$$

$$\displaystyle \phi_{0} = x y - x - y + 1$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{1}:v\mapsto v(1,0)$$

$$\displaystyle \phi_{1} = x \left(1 - y\right)$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{2}:v\mapsto v(0,1)$$

$$\displaystyle \phi_{2} = y \left(1 - x\right)$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{3}:v\mapsto v(1,1)$$

$$\displaystyle \phi_{3} = x y$$

This DOF is associated with vertex 3 of the reference element.
$$\displaystyle l_{4}:\mathbf{v}\mapsto\displaystyle\int_{e_{0}}(\sqrt{3} \cdot \left(60 s_{0}^{2} - 60 s_{0} + 10\right))v$$
where $$e_{0}$$ is the 0th edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{0}$$.

$$\displaystyle \phi_{4} = \sqrt{3} x \left(- x y + x + y - 1\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{5}:\mathbf{v}\mapsto\displaystyle\int_{e_{1}}(\sqrt{3} \cdot \left(60 s_{0}^{2} - 60 s_{0} + 10\right))v$$
where $$e_{1}$$ is the 1st edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{1}$$.

$$\displaystyle \phi_{5} = \sqrt{3} y \left(- x y + x + y - 1\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{6}:\mathbf{v}\mapsto\displaystyle\int_{e_{2}}(\sqrt{3} \cdot \left(60 s_{0}^{2} - 60 s_{0} + 10\right))v$$
where $$e_{2}$$ is the 2nd edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{2}$$.

$$\displaystyle \phi_{6} = \sqrt{3} x y \left(y - 1\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{7}:\mathbf{v}\mapsto\displaystyle\int_{e_{3}}(\sqrt{3} \cdot \left(60 s_{0}^{2} - 60 s_{0} + 10\right))v$$
where $$e_{3}$$ is the 3rd edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{3}$$.

$$\displaystyle \phi_{7} = \sqrt{3} x y \left(x - 1\right)$$

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{8}:\mathbf{v}\mapsto\displaystyle\int_{R}(10800 s_{0}^{2} s_{1}^{2} - 10800 s_{0}^{2} s_{1} + 1800 s_{0}^{2} - 10800 s_{0} s_{1}^{2} + 10800 s_{0} s_{1} - 1800 s_{0} + 1800 s_{1}^{2} - 1800 s_{1} + 300)v$$
where $$R$$ is the reference element;
and $$s_{0},s_{1}$$ is a parametrisation of $$R$$.

$$\displaystyle \phi_{8} = 3 x y \left(x y - x - y + 1\right)$$

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