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Asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes. Part 1: A smooth problem and globally quasi-uniform meshes

Authors: W. Hoffmann, A. H. Schatz, L. B. Wahlbin and G. Wittum
Journal: Math. Comp. 70 (2001), 897-909
MSC (2000): Primary 65N30, 65N15
Published electronically: March 7, 2001
MathSciNet review: 1826572
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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 averaging operators which are approximate gradients, ``recovered gradients'', which are then compared to the actual gradient of the approximation on each element. Conditions are given under which they are asympotically exact or equivalent estimators on each single element of the underlying meshes. Asymptotic exactness is accomplished by letting the approximate gradient operator average over domains that are large, in a controlled fashion to be detailed below, compared to the size of the elements.

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Additional Information

W. Hoffmann
Affiliation: ICA 3, Universität Stuttgart, Pfaffenwaldring 27, Stuttgart, Germany

A. H. Schatz
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853

L. B. Wahlbin
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853

G. Wittum
Affiliation: IWR, Universität Heidelberg, in Neuenheimer Feld 368, Heidelberg, Germany

Received by editor(s): November 20, 1998
Published electronically: March 7, 2001
Additional Notes: The second and third authors were supported by the National Science Foundation, USA
Article copyright: © Copyright 2001 American Mathematical Society

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