Boundary layer tails in periodic homogenization

Free access article

Issue		ESAIM: COCV Volume 4, 1999


Page(s)		209 - 243
DOI		10.1051/cocv:1999110

DOI: 10.1051/cocv:1999110

ESAIM: COCV, May 1999, Vol. 4, p. 209-243

Boundary layer tails in periodic homogenization

Queues de couches limites en homogénéisation périodique

Grégoire Allaire
Laboratoire d'Analyse Numérique, Université Pierre et Marie Curie, Paris VI, Tour 55-65, 4 place Jussieu, 70005 Paris, France; (allaire@ann.jussieu.fr)

Micol Amar
Dipartimento Me.Mo.Mat, Università "La Sapienza" - Roma, Via A. Scarpa 16, 00161 Roma, Italy; (amar@dmmm.uniroma1.it)

Received August 31, 1998. Revised January 21, 1999.

Abstract: This paper focus on the properties of boundary layers in periodic homogenization in rectangular domains which are either fixed or have an oscillating boundary. Such boundary layers are highly oscillating near the boundary and decay exponentially fast in the interior to a non-zero limit that we call boundary layer tail. The influence of these boundary layer tails on interior error estimates is emphasized. They mainly have two effects (at first order with respect to the period ${\varepsilon}$ ): first, they add a dispersive term to the homogenized equation, and second, they yield an effective Fourier boundary condition.

Résumé: Cet article est consacré à l'étude des couches limites en homogénéisation périodique dans des domaines rectangulaires qui, soit sont fixes, soit possèdent une frontière oscillante. Ces couches limites sont très oscillantes près du bord et décroissent exponentiellement vite à l'intérieur vers une limite non nulle que nous appelons queue de couche limite. L'influence de ces queues de couches limites pour l'obtention d'estimations d'erreur intérieures est mise à jour. Elles ont pour effets principaux (au premier ordre en ${\varepsilon}$ , la période) : premièrement, d'ajouter un terme dispersif dans l'équation homogénéisée, deuxièmement, de conduire à une condition aux limites effective de type Fourier.

Keywords and phrases: Boundary layers, periodic functions, asymptotic expansion, homogenization.

AMS Subject Classification: 35B27

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