The Comparison of Two Error Estimates for Approximate Solutions of the Poisson Equation

Abstract

The purpose of the present paper is to discuss different error estimates for the numerical solution of a Dirichlet problem for the Poisson equation, calculated via the five point (discrete) Laplacian. Whereas the first error bound, a sum of ordinary $L^{\infty }$-moduli, is deduced from the usual stability inequality, use is made of some properties of the discrete Green function to verify another stability inequality in terms of $l^1$-norms, which then implies the second estimate via $\tau$-moduli multiplied by a logarithm factor. In the following we will show that it depends on the solution of the boundary value problem which measure of smoothness or rather which error estimate delivers the correct rate of convergence. The paper concludes with some remarks illustrating relations to the well-known logarithm factor in connection with finite element approximation.