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

This paper devotes to the construction of finite elements on grids that consist of general quadrilaterals not limited in parallelograms. Two finite elements are established for the H^1 and H( rot) elliptic problems, respectively. The two finite element spaces on general quadrilateral grids, together with the space of piecewise constant functions, formulate a discretised de Rham complex. First order convergence rate can be proved for both of them under the asymptotic-parallelogram assumption on the grids. The local shape functions of the two spaces are piecewise polynomials, and the global finite element functions are nonconforming. A rigorous analysis is given in this paper that it is impossible to construct a practically useful finite element defined as Ciarlet's triple whose shape functions are always piecewise polynomials and which can form conforming subspaces on a grid that consists of arbitrary quadrilaterals.

READ FULL TEXT