Superconvergence of the Shortley–Weller approximation for Dirichlet problems

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⊂R2 with the boundary Γ. This means that if the exact solution belongs to C3,1(Ω̄), then the approximate solution obtained by the S–W formula gives O(h3) accuracy at every grid point whose distance to Γ is O(h) and O(h2) 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∈Cl+2,α(Ω̄), where l=0 or 1 and α∈(0,1) stands for the Hölder exponent. Numerical examples are also given, which illustrate our results.