Asymptotic expansion of steady-state potential in a high contrast medium with a thin resistive layer

We study the steady-state potential in a high contrast medium with a resistive thin layer. We provide asymptotic expansion of the potential at any order. Transmission conditions at any order and the corresponding variational formulation are given. We prove uniform estimates with respect to the thickness of the layer and with respect to its resistivity. The main insight consists in the uniform variational formulation whatever small the layer conductivity is. Numerical simulations illustrate the theoretical results. Key-words: Asymptotic analysis, Finite Element Method, Laplace equations ∗ 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 42 65 9, v er si on 2 7 Ju n 20 11 Conditions de transmission approchées pour des couches minces résistives. Résumé : Mots-clés : Analyse Asymptotique, Méthode des Eléments Finis, Equations de Laplace in ria -0 04 42 65 9, v er si on 2 7 Ju n 20 11 Transmission conditions for resistive thin layers 3