Title:High order transmission conditions for the homogenization of interface problems
Abstract: This work is a joint work with Xavier Claeys (UPMC, University Paris 6) and Valentin Vinoles (former PhD student, now at EPFL). Classical homogenization theory takes into account poorly interfaces (or boundaries). Indeed, it is well known that the presence of interfaces (as boundaries) induces a loss of accuracy. This is linked to the presence of boundary layers near the interfaces. The objective of this work is to construct approximate effective transmission conditions that would enrich the model near the interfaces and restore the desired accuracy.
We have first considered a plane interface between a homogeneous and a periodic media in the standard case. Coupling a classical two-scale expansion with matched asymptotic expansion, we obtain high order transmission conditions between the homogeneous media and the homogenized media. Those conditions are non standard : they involve Laplace-Beltrami operators at the interface and requires in particular to solve non classical cell problems in infinite partly periodic strips. We have introduced an approximate model for which stability properties have been proven. Error estimates justify that this new model is more accurate than the classical one near the interface and in the bulk. Numerical results, obtained using Dirichlet-to-Neumann operators in periodic media, validates the theoretical results.
