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Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr


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Volume 17, Issue 3

Issues

A Posteriori Modelling-Discretization Error Estimate for Elliptic Problems with L -Coefficients

Monika Weymuth / Stefan Sauter / Sergey Repin
  • Corresponding author
  • Russian Academy of Sciences, Saint Petersburg Department of V. A. Steklov Institute of Mathematics, Fontanka 27, 191 011 Saint Petersburg, Russia; and University of Jyväskylä, P.O. Box 35, FI-40014, Jyväskylä, Finland
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Published Online: 2017-06-21 | DOI: https://doi.org/10.1515/cmam-2017-0015

Abstract

We consider elliptic problems with complicated, discontinuous diffusion tensor A0. One of the standard approaches to numerically treat such problems is to simplify the coefficient by some approximation, say Aε, and to use standard finite elements. In [19] a combined modelling-discretization strategy has been proposed which estimates the discretization and modelling errors by a posteriori estimates of functional type. This strategy allows to balance these two errors in a problem adapted way. However, the estimate of the modelling error was derived under the assumption that the difference A0-Aε becomes small with respect to the L-norm. This implies in particular that interfaces/discontinuities separating the smooth parts of A0 have to be matched exactly by the coefficient Aε. Therefore the efficient application of that theory to problems with complicated or curved interfaces is limited. In this paper, we will present a refined theory, where the difference A0-Aε is measured in the Lq-norm for some appropriate q]2,[ and, hence, the geometric resolution condition is significantly relaxed.

Keywords: A Posteriori Error Estimation; Elliptic Regularity; Model Simplification

MSC 2010: 65N30; 65N15; 35J25

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About the article

Received: 2017-04-06

Revised: 2017-05-18

Accepted: 2017-05-19

Published Online: 2017-06-21

Published in Print: 2017-07-01


Citation Information: Computational Methods in Applied Mathematics, Volume 17, Issue 3, Pages 515–531, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2017-0015.

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