Two-scale analysis for very rough thin layers. An explicit characterization of the polarization tensor

We study the behaviour of the steady-state voltage potential in a material composed of a two-dimensional object surrounded by a very rough thin layer and embedded in an ambient medium. The roughness of the layer is described by a quasi ε–periodic function, ε being a small parameter, while the mean thickness of the layer is of magnitude ε, where β ∈ (0, 1). Using the twoscale analysis, we replace the very rough thin layer by appropriate transmission conditions on the boundary of the object, which lead to an explicit characterization of the polarization tensor of Vogelius and Capdeboscq (ESAIM:M2AN. 2003; 37:159-173). This paper extends the previous works Poignard (Math. Meth. App. Sci. 2009; 32:435-453) and Ciuperca et al. (Research report INRIA RR-6812), in which β ≥ 1. Key-words: Asymptotic analysis, Finite Element Method, Laplace equations ∗ Université de Lyon, Université Lyon 1, CNRS, UMR 5208, Institut Camille Jordan, Bat. Braconnier, 43 boulevard du 11 novembre 1918, F 69622 Villeurbanne Cedex, France † Laboratoire Ampère UMR CNRS 5005, Universit de Lyon, École Centrale de Lyon, F-69134 Écully, France ‡ INRIA Bordeaux-Sud-Ouest, Institut de Mathématiques de Bordeaux, CNRS UMR 5251 & Université de Bordeaux1, 351 cours de la Libération, 33405 Talence Cedex, France in ria -0 04 01 83 5, v er si on 1 6 Ju l 2 00 9 Analyse double échelle pour des couches minces très rugueuses. Une caractérisation explicite du tenseur de polarisation Résumé : Mots-clés : Analyse Asymptotique, Méthode des Eléments Finis, Equations de Laplace in ria -0 04 01 83 5, v er si on 1 6 Ju l 2 00 9 Approximate transmission conditions for very rough thin layers 3