Alfred H. Schatz

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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)
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)
A class of a posteriori estimators is studied for the error in the maximum-norm of the gradient on single elements when the finite element method is used to approximate solutions of second order elliptic problems. The meshes are unstructured and, in particular, it is not assumed that there are any known superconvergent points. The estimators are based on(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)
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)
We extend results from Part I about estimating gradient errors elementwise a posteriori, given there for quadratic and higher elements, to the piecewise linear case. The key to our new result is to consider certain technical estimates for differences in the error, e(x 1) − e(x 2), rather than for e(x) itself. We also give a posteriori estimators for second(More)
We consider piecewise linear finite element approximations uh to u the solution of an elliptic boundary value problem. New estimates for the differences |e(x1) − e(x2)| (where e(x) = u(x) − uh(x) is the error and x1 and x2 are any two points in the domain) are obtained in terms of weighted L∞ norms. As a consequence, so-called asymptotic expansion(More)