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