Maximum-norm interior estimates for Ritz-Galerkin methods

Abstract

In this paper we obtain, by simple means, interior maximum-norm estimates for a class of Ritz-Galerkin methods used for approximating solutions of second order elliptic boundary value problems in R N {{\mathbf {R}}^N} . The estimates are proved when the approximating subspaces are any of a large class of piecewise polynomial subspaces which we assume here to be defined on a uniform mesh on the interior domain. Optimal rates of convergence are obtained.