Comptes Rendus
no. 23-24
Mathematical analysis/Numerical analysis
A posteriori error majorants of the modeling errors for elliptic homogenization problems
[Majorations a posteriori de lʼerreur de modélisation de problèmes elliptiques homogénéisés]

Présenté par : Philippe G. Ciarlet

Sergey I. Repin1 ; Tatiana S. Samrowski2 ; Stefan A. Sauter3
1 University of Jyväskylä, Mattilanniemi 2, 40100 Jyväskylä, Finland
2 School of Engineering, Technikumstrasse 9, 8400 Winterthur, Switzerland
3 Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zurich, Switzerland
Comptes Rendus. Mathématique, Volume 351 (2013) no. 23-24, pp. 877-882.

Dans cette Note, nous obtenons de nouvelles estimations de lʼerreur de modélisation pour des problèmes elliptiques linéaires dʼhomogénéisation à coefficients périodiques. Notre approche est fondée sur le concept dʼestimation a posteriori fonctionnelle. Nos estimations sont obtenues pour la norme dʼénergie et utilisent seulement le flux de la solution non oscillante du problème homogénéisé et la solution dʼun problème aux limites sur la cellule de périodicité.

In this paper, we derive new two-sided a posteriori estimates of the modeling errors for linear elliptic boundary value problems with periodic coefficients solved by homogenization. Our approach is based on the concept of functional a posteriori error estimation. The estimates are obtained for the energy norm and use solely the global flux of the non-oscillatory solution of the homogenized model and solution of a boundary value problem on the cell of periodicity.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2013.10.025
Sergey I. Repin 1 ; Tatiana S. Samrowski 2 ; Stefan A. Sauter 3

1 University of Jyväskylä, Mattilanniemi 2, 40100 Jyväskylä, Finland
2 School of Engineering, Technikumstrasse 9, 8400 Winterthur, Switzerland
3 Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zurich, Switzerland 
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     title = {A posteriori error majorants of the modeling errors for elliptic homogenization problems},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {877--882},
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Sergey I. Repin; Tatiana S. Samrowski; Stefan A. Sauter. A posteriori error majorants of the modeling errors for elliptic homogenization problems. Comptes Rendus. Mathématique, Volume 351 (2013) no. 23-24, pp. 877-882. doi : 10.1016/j.crma.2013.10.025. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.10.025/

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[5] S.I. Repin; T. Samrowski; S. Sauter Combined a posteriori modelling-discretization error eestimate for elliptic problems with variable coefficients, ESAIM: Math. Model. Numer. Anal., Volume 46 (2012), pp. 1389-1405

[6] S.I. Repin; T. Samrowski; S. Sauter Two-sided estimates of the modeling error for elliptic homogenization problems, Institut für Mathematik, Univ. Zürich, 2012 http://www.math.uzh.ch/compmath/fileadmin/math/preprints/12_12.pdf (Technical Report 12-2012)

[7] S. Repin; S. Sauter; A. Smolianski A posteriori error estimation for the Dirichlet problem with account of the error in boundary conditions, Computing, Volume 70 (2003), pp. 205-233

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