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A posteriori error estimation for the Dirichlet problem with account of the error in the approximation of boundary conditions


Repin, S; Sauter, S; Smolianski, A (2003). A posteriori error estimation for the Dirichlet problem with account of the error in the approximation of boundary conditions. Computing, 70(3):205-233.

Abstract

The present work is devoted to the a posteriori error estimation for 2nd order elliptic problems with Dirichlet boundary conditions. Using the duality technique we derive the reliable and efficient a posteriori error estimator that measures the error in the energy norm. The estimator can be used in assessing the error of any approximate solution which belongs to the Sobolev space H1, independently of the discretization method chosen. In particular, our error estimator can be applied also to problems and discretizations where the Galerkin orthogonality is not available. We will present different strategies for the evaluation of the error estimator. Only one constant appears in its definition which is the one from Friedrichs' inequality; that constant depends solely on the domain geometry, and the estimator is quite non-sensitive to the error in the constant evaluation. Finally, we show how accurately the estimator captures the local error distribution, thus, creating a base for a justified adaptivity of an approximation.

The present work is devoted to the a posteriori error estimation for 2nd order elliptic problems with Dirichlet boundary conditions. Using the duality technique we derive the reliable and efficient a posteriori error estimator that measures the error in the energy norm. The estimator can be used in assessing the error of any approximate solution which belongs to the Sobolev space H1, independently of the discretization method chosen. In particular, our error estimator can be applied also to problems and discretizations where the Galerkin orthogonality is not available. We will present different strategies for the evaluation of the error estimator. Only one constant appears in its definition which is the one from Friedrichs' inequality; that constant depends solely on the domain geometry, and the estimator is quite non-sensitive to the error in the constant evaluation. Finally, we show how accurately the estimator captures the local error distribution, thus, creating a base for a justified adaptivity of an approximation.

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32 citations in Web of Science®
30 citations in Scopus®
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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:A posteriori error estimate; duality technique; reliability; efficiency; local error distribution
Language:English
Date:2003
Deposited On:29 Nov 2010 16:26
Last Modified:05 Apr 2016 13:25
Publisher:Springer
ISSN:0010-485X
Additional Information:The original publication is available at www.springerlink.com
Publisher DOI:https://doi.org/10.1007/s00607-003-0013-7A
Related URLs:http://www.zentralblatt-math.org/zbmath/search/?q=an%3A1128.35319
http://www.ams.org/mathscinet-getitem?mr=2011610
Permanent URL: https://doi.org/10.5167/uzh-21896

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