Enriched homogenization in presence of boundaries or interfaces

This work is motivated by the fact that classical homogenization theory poorly takes into account interfaces or boundaries. It is particularly unfortunate when one is interested in phenomena arising at the interfaces or the boundaries of the periodic media (the propagation of plasmonic waves at the surface of metamaterials for instance). To overcome this limitation, we have constructed an effective model which is enriched near the interfaces and the boundaries. For now, we have treated and analysed the case of simple geometries : for instance a plane interface between a homogeneous and a periodic half spaces. We have derived a high order approximate model which consists in replacing the periodic media by an effective one but the transmission conditions are not classical. The obtained conditions involve Laplace- Beltrami operators at the interface and requires to solve cell problems in periodicity cell (as in classical homogenization) and in infinite strips (to take into account the phenomena near the interface). We establish well posedness for the approximate model and error estimates which justify that this new model is more accurate near the interface and in the bulk. This will be illustrated by numerical results.