Corrector estimates in homogenization of a nonlinear transmission problem for diffusion equations in connected domains

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. KEYWORDS bidomain model, corrector estimates, diffusion problem, nonlinear transmission conditions, periodic unfolding technique MSC CLASSIFICATION 35B27; 35M10; 82C24This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

Existence and two-scale convergence of the generalised Poisson–Nernst–Planck problem with non-linear interface conditions

“…We note that the scaling with γ 0 yields no contribution of the interface reaction term in the macroscopic model (3.4a). For a possible remedy, in [25], we investigate the bi-domain setting of a non-linear transmission problem for the linear diffusion equation in connected domains.…”

“…• Based on the two-scale convergence and using the scale transformation between twophase domains, we rigorously prove a new homogenisation result for the doubly non-linear drift-diffusion system of PNP focusing on the inhomogeneous flux interface conditions. Compared to the other own works [25,29], we rely on convergences without residual error estimates for the sharp scaling of the interface fluxes with ε.…”

Inhomogeneous Boundary Value Problems for the Generalized Boussinesq Model of Mass Transfer

“…Finally, we mention papers [45][46][47][48][49][50][51][52][53] that touch upon issues close to the subject of this paper from nonlinear diffusion, viscoelasticity, engineering mechanics, applied heat transfer and magnetic hydrodynamics, acoustics, and oceanology.…”