Degree 0 Arnold–Boffi–Falk on a quadrilateral

• $R$ 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:

${\displaystyle l_0:\mathit{v}\mapsto {\displaystyle {\int }_{e_0}\mathit{v}\cdot (1){\hat{\mathit{n}}}_0}}$
where $e_0$ is the 0th edge;
and ${\hat{\mathit{n}}}_0$ is the normal to facet 0.

${\displaystyle {\mathit{\varphi }}_0=\left(\begin{array}{c}{\displaystyle 3x(x-1)}\\ {\displaystyle 3y^2-4y+1}\end{array}\right)}$

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

${\displaystyle l_1:\mathit{v}\mapsto {\displaystyle {\int }_{e_1}\mathit{v}\cdot (1){\hat{\mathit{n}}}_1}}$
where $e_1$ is the 1st edge;
and ${\hat{\mathit{n}}}_1$ is the normal to facet 1.

${\displaystyle {\mathit{\varphi }}_1=\left(\begin{array}{c}{\displaystyle -3x^2+4x-1}\\ {\displaystyle 3y(1-y)}\end{array}\right)}$

This DOF is associated with edge 1 of the reference element.

${\displaystyle l_2:\mathit{v}\mapsto {\displaystyle {\int }_{e_2}\mathit{v}\cdot (1){\hat{\mathit{n}}}_2}}$
where $e_2$ is the 2nd edge;
and ${\hat{\mathit{n}}}_2$ is the normal to facet 2.

${\displaystyle {\mathit{\varphi }}_2=\left(\begin{array}{c}{\displaystyle x(3x-4)}\\ {\displaystyle 3y(y-1)}\end{array}\right)}$

This DOF is associated with edge 2 of the reference element.

${\displaystyle l_3:\mathit{v}\mapsto {\displaystyle {\int }_{e_3}\mathit{v}\cdot (1){\hat{\mathit{n}}}_3}}$
where $e_3$ is the 3rd edge;
and ${\hat{\mathit{n}}}_3$ is the normal to facet 3.

${\displaystyle {\mathit{\varphi }}_3=\left(\begin{array}{c}{\displaystyle 3x(1-x)}\\ {\displaystyle y(4-3y)}\end{array}\right)}$

This DOF is associated with edge 3 of the reference element.

${\displaystyle l_4:\mathbf{v}\mapsto {\displaystyle {\int }_R(s_0)\mathrm{\nabla }\cdot \mathbf{v}}}$
where $R$ is the reference element;
and $s_0,s_1$ is a parametrisation of $R$.

${\displaystyle {\mathit{\varphi }}_4=\left(\begin{array}{c}{\displaystyle 6x(x-1)}\\ {\displaystyle 0}\end{array}\right)}$

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

${\displaystyle l_5:\mathbf{v}\mapsto {\displaystyle {\int }_R(s_1)\mathrm{\nabla }\cdot \mathbf{v}}}$
where $R$ is the reference element;
and $s_0,s_1$ is a parametrisation of $R$.

${\displaystyle {\mathit{\varphi }}_5=\left(\begin{array}{c}{\displaystyle 0}\\ {\displaystyle 6y(y-1)}\end{array}\right)}$

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