Hölder estimates for Green’s functions on convex polyhedral domains and their applications to finite element methods

Abstract

A model second-order elliptic equation on a general convex polyhedral domain in three dimensions is considered. The aim of this paper is twofold: First sharp Holder estimates for the corresponding Green’s function are obtained. As an applications of these estimates to finite element methods, we show the best approximation property of the error in $${W^1_{\infty}}$$. In contrast to previously known results, $${W_p^{2}}$$regularity for p > 3, which does not hold for general convex polyhedral domains, is not required. Furthermore, the new Green’s function estimates allow us to obtain localized error estimates at a point.