A sharp error estimate for the numerical solution of multivariate Dirichlet problems

For the multidimensional Dirichlet problem of the Poisson equation on an arbitrary compact domain, this study examines convergence properties with rates of approximate solutions, obtained by a standard difference scheme over inscribed uniform grids. Sharp quantitative estimates are given by the use of second moduli of continuity of the second single partial derivatives of the exact solution. This is achieved by employing the probabilistic method of simple random walk.