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)


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”.


Porous Medium Single Phase Flow Homogeneous Boundary Condition Homogenization Theory Limit Solution 
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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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