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Journal of Numerical Mathematics

Editor-in-Chief: Hoppe, Ronald H. W. / Kuznetsov, Yuri

Managing Editor: Olshanskii, Maxim

Editorial Board: Benzi, Michele / Brenner, Susanne C. / Carstensen, Carsten / Dryja, M. / Feistauer, Miloslav / Glowinski, R. / Lazarov, Raytcho / Nataf, Frédéric / Neittaanmaki, P. / Bonito, Andrea / Quarteroni, Alfio / Guzman, Johnny / Rannacher, Rolf / Repin, Sergey I. / Shi, Zhong-ci / Tyrtyshnikov, Eugene E. / Zou, Jun / Simoncini, Valeria / Reusken, Arnold

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Volume 24, Issue 1


Estimates of the modeling error generated by homogenization of an elliptic boundary value problem

Sergey Repin
  • Corresponding author
  • V. A. Steklov Institute of Mathematics, Fontanka 27, 191011 St. Petersburg, Russia, and University of Jyväskylä, Finland
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/ Tatiana Samrowski
  • Institute for Applied Mathematics and Physics, Zurich University of Applied Sciences, Technikumstrasse 6, CH-8400 Winterthur, Switzerland
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/ Stefan Sauter
  • Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zurich, Switzerland
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Published Online: 2016-03-14 | DOI: https://doi.org/10.1515/jnma-2014-1002


In this paper, we derive a posteriori bounds of the difference between the exact solution of an elliptic boundary value problem with periodic coefficients and an abridged model, which follows from the homogenization theory. The difference is measured in terms of the energy norm of the basic problem and also in the combined primal–dual norm. Using the technique of functional type a posteriori error estimates, we obtain two-sided bounds of the modelling error, which depends only on known data and the solution of the homogenized problem. It is proved that the majorant with properly chosen arguments possesses the same convergence rate, which was established for the true error. Numerical tests confirm the efficiency of the estimates.

Keywords: periodic structures; homogenization; elliptic boundary value problems; a posteriori error estimates; modeling error

MSC 2010: 35J15; 35B27; 65N15


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About the article

Received: 2014-01-26

Revised: 2014-11-21

Accepted: 2015-01-28

Published Online: 2016-03-14

Published in Print: 2016-03-01

Citation Information: Journal of Numerical Mathematics, Volume 24, Issue 1, Pages 1–15, ISSN (Online) 1569-3953, ISSN (Print) 1570-2820, DOI: https://doi.org/10.1515/jnma-2014-1002.

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