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Homogenization in a Perforated Domain Including a Thin Full Interlayer

  • Alain Bourgeat
  • Roland Tapiéro
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 114)

Abstract

We study the behavior of the solution of a model elliptic problem, the Poisson equation −Δu ε = f on a domain D ε including many tiny holes. These holes are periodically distributed with a period diameter equal to ɛ except in a thin, unperforated layer of thickness η crossing the domain. We show that, if ɛ, η tend to zero and the ratio η/ɛ tends to zero or a constant, the behavior of the limit solution is just as if there were no unperforated layer and only a whole, periodically perforated domain. If \(\varepsilon < \eta \leqslant {\varepsilon ^{\tfrac{2}{3}}}\), the layer has an influence only on itself but not on the perforated parts. If \(\eta > {\varepsilon ^{\tfrac{2}{3}}}\), the asymptotic behavior of the solution is different everywhere in D ɛ from the classical solution of Lions [10] in perforated domains. Proofs are given using energy estimates in Sobolev spaces and the framework of homogenization theory; limits can be found by the classical techniques using “test functions” and “two-scale convergence”.

Keywords

Porous Medium Single Phase Flow Homogeneous Boundary Condition Homogenization Theory Limit Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Basel AG 1993

Authors and Affiliations

  • Alain Bourgeat
    • 1
  • Roland Tapiéro
    • 2
  1. 1.URA CNRS 0740, Equipe d’Analyse NumériqueUniversité de Saint-EtienneSaint Etienne Cedex 2France
  2. 2.URA CNRS 0740, Laboratoire d’Analyse NumériqueUniversité Claude Bernard Lyon IVilleurbanne CedexFrance

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