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Corrector estimates for the homogenization of a locally periodic medium with areas of low and high diffusivity

Published online by Cambridge University Press:  02 April 2013

A. MUNTEAN  and
T. L. VAN NOORDEN
A. MUNTEAN
Affiliation:
CASA – Centre for Analysis, Scientific computing and Applications, Department of Mathematics and Computer Science, Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands email: [email protected] Institute for Complex Molecular Systems (ICMS), Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
T. L. VAN NOORDEN
Affiliation:
Chair of Applied Mathematics 1, Department of Mathematics, University of Erlangen-Nürnberg, Martensstraße 3, Erlangen 91058, Germany
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Abstract

We prove an upper bound for the convergence rate of the homogenization limit ε → 0 for a linear transmission problem for a advection–diffusion(–reaction) system posed in areas with low and high diffusivity, where ε is a suitable scale parameter. In this way we rigorously justify the formal homogenization asymptotics obtained in [37] (van Noorden, T. and Muntean, A. (2011) Homogenization of a locally-periodic medium with areas of low and high diffusivity. Eur. J. Appl. Math. 22, 493–516). We do this by providing a corrector estimate. The main ingredients for the proof of the correctors include integral estimates for rapidly oscillating functions with prescribed average, properties of the macroscopic reconstruction operators, energy bounds, and extra two-scale regularity estimates. The whole procedure essentially relies on a good understanding of the analysis of the limit two-scale problem.

Keywords

Corrector estimatesTransmission conditionHomogenizationMicro–macro transportReaction–diffusion system in heterogeneous materials
Type
Papers
Information
European Journal of Applied Mathematics , Volume 24 , Issue 5 , October 2013 , pp. 657 - 677
Copyright
Copyright © Cambridge University Press 2013 

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