Abstract

This paper is devoted to the homogenization of a nonlinear transmission problem stated in a two‐phase domain. We consider a system of linear diffusion equations defined in a periodic domain consisting of two disjoint phases that are both connected sets separated by a thin interface. Depending on the field variables, at the interface, nonlinear conditions are imposed to describe interface reactions. In the variational setting of the problem, we prove the homogenization theorem and a bidomain averaged model. The periodic unfolding technique is used to obtain the residual error estimate with a first‐order corrector.

Highlights

  • We consider coupled linear parabolic equations describing the diffusion of two species in two different phases of one physical domain separated by a thin periodic interface

  • The coupling of the species arises via nonlinear transmission conditions at the interface, which model surface reactions

  • The limit bidomain model is given via two coupled parabolic equations defined in the macroscopic domain describing the diffusion of the two species in each phase and reactions at the interface

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Summary

INTRODUCTION

We consider coupled linear parabolic equations describing the diffusion of two species in two different phases of one physical domain separated by a thin periodic interface. Unfolding-based error estimates have been proven for linear, elliptic transmission problems in Reichelt,[26] for reaction-diffusion systems with linear boundary conditions in perforated domains in Muntean and Reichelt,[27] and for systems with nonlinear interface conditions in a two-phase domain in Fatima et al.[28] The latter results are based on the quantification of the periodicity defect for the periodic unfolding operator in Griso,[29,30] and they hold without assuming higher regularity for the corrector problem.

SETTING OF THE TRANSMISSION PROBLEM
WELL-POSEDNESS
PERIODIC UNFOLDING TECHNIQUE
ASYMPTOTIC ANALYSIS
THE MAIN HOMOGENIZATION RESULT
DISCUSSION

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