Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids: Part I. Global estimates

This part contains new pointwise error estimates for the finite element method for second order elliptic boundary value problems on smooth bounded domains in RN . In a sense to be discussed below these sharpen known quasi–optimal L∞ and W 1 ∞ estimates for the error on irregular quasi–uniform meshes in that they indicate a more local dependence of the error at a point on the derivatives of the solution u. We note that in general the higher order finite element spaces exhibit more local behavior than lower order spaces. As a consequence of these estimates new types of error expansions will be derived which are in the form of inequalities. These expansion inequalities are valid for large classes of finite elements defined on irregular grids in RN and have applications to superconvergence and extrapolation and a posteriori estimates. Part II of this series will contain local estimates applicable to non–smooth problems. 0. Introduction and discussion of results This is the first of a series of papers whose aim is to derive new pointwise error estimates for the finite element method on general quasi–uniform meshes for second order elliptic boundary value problems in R , N ≥ 2. In a sense to be discussed below, these estimates represent an improvement on the now standard quasi–optimal L∞ estimates. In order to fix the ideas, here we will deal with global estimates for a model Neumann problem with smooth solutions. In succeeding papers, local estimates, both interior and up to the boundary, which are applicable to a variety of problems with both smooth and nonsmooth solutions will be considered. As a consequence of these estimates, some new and useful inequalities will be given which are in the form of error expansions. They are valid for large classes of finite elements on general quasi–uniform meshes in R and have application to superconvergence and extrapolation and a posteriori esitmates. Let us begin by giving a brief description of some of the main results of this paper. Let Ω be a bounded domain in R , N ≥ 2, with smooth boundary ∂Ω. Let