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This part contains new pointwise error estimates for the finite element method for second order elliptic boundary value problems on smooth bounded domains in R N. 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… (More)

New uniform error estimates are established for finite element approximations u h of solutions u of second-order elliptic equations Lu = f using only the regularity assumption u 1 ≤ cf −1. Using an Aubin–Nitsche type duality argument we show for example that, for arbitrary (fixed) ε sufficiently small, there exists an h 0 such that for 0 < h < h 0 u − u h 0… (More)

A model second-order elliptic equation on a general convex poly-hedral domain in three dimensions is considered. The aim of this paper is twofold: First sharp Hölder 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 ∞. In… (More)

Local energy error estimates for the finite element method for el-liptic problems were originally proved in 1974 by Nitsche and Schatz. These estimates show that the local energy error may be bounded by a local approximation term, plus a global " pollution " term that measures the influence of solution quality from outside the domain of interest and is… (More)

This paper presents a nonstandard local approach to Richardson extrapolation, when it is used to increase the accuracy of the standard finite element approximation of solutions of second order elliptic boundary value problems in R N , N ≥ 2. The main feature of the approach is that it does not rely on a traditional asymptotic error expansion, but rather… (More)

We consider finite element methods for a model second-order el-liptic equation on a general bounded convex polygonal or polyhedral domain. Our first main goal is to extend the best approximation property of the error in the í µí± 1 ∞ norm, which is known to hold on quasi-uniform meshes, to more general graded meshes. We accomplish it by a novel proof… (More)

Our aim here is to give sufficient conditions on the finite element spaces near a point so that the error in the finite element method for the function and its derivatives at the point have exact asymptotic expansions in terms of the mesh parameter h, valid for h sufficiently small. Such expansions are obtained from the so-called asymptotic expansion… (More)