Convergence of consistent and inconsistent finite difference schemes and an acceleration technique

This paper states and generalizes in part some recent results on finite difference methods for Dirichlet problems in a bounded domain Ω which the author has obtained by himself or with coworkers. After stating a superconvergence property of finite difference solution for the case where the exact solution u belongs to C4(Ω̄), it is remarked that such a property does not hold in general if u∉C4(Ω̄). Next, a convergence theorem is given for inconsistent schemes under some assumptions. Furthermore, it is shown that the accuracy of the approximate solution can be improved by a coordinate transformation. Numerical examples are also given.