A lowest-degree quasi-conforming finite element de Rham complex on general quadrilateral grids by piecewise polynomials

This paper discusses finite elements defined by Ciarlet’s triple on grids that consist of general quadrilaterals not limited in parallelograms. Specifically, two finite elements are established for the H¹ and H(r ot) elliptic problems, respectively. An O(h) order convergence rate in energy norm for both of them and an O(h²) order convergence in L² norm for the H¹ scheme are proved under the O(h²) asymptotic-parallelogram assumption on the grids. Further, these two finite element spaces, together with the space of piecewise constant functions, formulate a discretized de Rham complex on general quadrilateral grids. 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 by 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.