H1‐superconvergence of finite difference method based on Q1‐element on quasi‐uniform mesh for the 3D Poisson equation

Abstract

AbstractIn this paper, the finite difference (FD) method is considered for the 3D Poisson equation by using the Q1‐element on a quasi‐uniform mesh. First, under the regularity assumption of , the H1‐superconvergence of the FD solution uh based on the Q1‐element to the first‐order interpolation function is obtained. Next, the H1‐superconvergence of the second‐order interpolation postprocessing function based on the FD solution uh to u is provided. Finally, numerical tests are presented to show the H1‐superconvergence result of the FD postprocessing function to u if .