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# Degree 0 Arnold–Boffi–Falk 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: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{5}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot\hat{\boldsymbol{n}}_{0}$$
where $$e_{0}$$ is the 0th edge;
and $$\hat{\boldsymbol{n}}_{0}$$ is the normal to facet 0.

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

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot\hat{\boldsymbol{n}}_{1}$$
where $$e_{1}$$ is the 1st edge;
and $$\hat{\boldsymbol{n}}_{1}$$ is the normal to facet 1.

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

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot\hat{\boldsymbol{n}}_{2}$$
where $$e_{2}$$ is the 2nd edge;
and $$\hat{\boldsymbol{n}}_{2}$$ is the normal to facet 2.

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

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot\hat{\boldsymbol{n}}_{3}$$
where $$e_{3}$$ is the 3rd edge;
and $$\hat{\boldsymbol{n}}_{3}$$ is the normal to facet 3.

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

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{4}:\mathbf{v}\mapsto\displaystyle\int_{R}(s_{0})\nabla\cdot\mathbf{v}$$
where $$R$$ is the reference element;
and $$s_{0},s_{1}$$ is a parametrisation of $$R$$.

$$\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 6 x \left(x - 1\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{5}:\mathbf{v}\mapsto\displaystyle\int_{R}(s_{1})\nabla\cdot\mathbf{v}$$
where $$R$$ is the reference element;
and $$s_{0},s_{1}$$ is a parametrisation of $$R$$.

$$\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 6 y \left(y - 1\right)\end{array}\right)$$

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