Superconvergence of the Shortley-Weller approximation for Dirichlet problems

摘要

This paper presents a superconvergence property of the Shortley-Weller (S-W) approximation applied to the Poisson-type Dirichlet problem in a bounded domain Ω is contained in R~2 with the boundary Γ. This means that if the exact solution belongs to C~(3,1)(Ω-bar), then the approximate solution obtained by the S-W formula gives O(h~3) accuracy at every grid point whose distance of Γ is O(h) and O(h~2) accuracy at other grid points, where h denotes the equal mesh-size in x and y directions. The similar property holds for the case u ∈ C~(l+2,α)(Ω-bar), where l = 0 or 1 and α ∈ (0,1) stands for the Holder exponent. Numerical examples are also given, which illustrate our results.