Lowest-degree piecewise polynomial de Rham complex on general quadrilateral grids

This paper is devoted to the construction of finite elements on grids that consist of general quadrilaterals not limited in parallelograms. Two finite elements defined as Ciarlet's triple are established for the $H^1$ and $H(\rm rot)$ elliptic problems, respectively. An $\mathcal{O}(h)$ order convergence rate in energy norm for both of them and an $\mathcal{O}(h^2)$ order convergence in $L^2$ norm for the $H^1$ scheme are proved under the asymptotic-parallelogram assumption on the grids. Further, the two finite element spaces on general quadrilateral grids, together with the space of piecewise constant functions, formulate a discretized de Rham complex. The finite element spaces consist of piecewise polynomial functions, and, thus, are nonconforming on general quadrilateral grids. Indeed, a rigorous analysis is given in this paper that it is impossible to construct a practically useful finite element defined as Ciarlet's triple that can formulate a finite element space which consists of continuous piecewise polynomial functions on a grid that may include arbitrary quadrilaterals.