Superconvergence of solution derivatives for the Shortley–Weller difference approximation of Poissonʹs equation. Part I: smoothness problems

The finite difference method (FDM) using the Shortley–Weller approximation can be viewed as a special kind of the finite element methods (FEMs) using the piecewise bilinear and linear functions, and involving some integration approximation. When u∈C3(S̄) (i.e., u∈C3,0(S̄)) and f∈C2(S̄), the superconvergence rate O(h2) of solution derivatives in discrete H1 norms by the FDM is derived for rectangular difference grids, where h is the maximal mesh length of difference grids used, and the difference grids are not confined to be quasiuniform. Comparisons are made on the analysis by the maximum principle and the FEM analysis, conversions between the FDM and the linear and bilinear FEMs are discussed, and numerical experiments are provided to support superconvergence analysis made.