Abstract

In this talk, we consider the 2D Helmholtz equation with coefficients that coincide with different periodic functions on both sides of a given interface. A numerical method has been proposed by [Fliss, Cassan, and Bernier] to solve this equation under the critical assumption that the overall medium stays periodic in the direction of the interface. In fact, in this case, a Floquet-Bloch transform can be applied with respect to the variable along the interface, thus leading to a family of closed waveguide problems. The purpose of this work is to deal with the case where the overall medium is no longer periodic in the direction of the interface (that is for instance if one of the half-spaces is not cut in a direction of periodicity, or if both half-spaces are periodic along the interface, but with incommensurate periods). As it is done in the works of [Gérard-Varet and Masmoudi] or [Blanc, Le Bris, and Lions], we use the crucial (but non-obvious) observation that the medium has a quasiperiodic structure along the interface, namely, it is the restriction of a higher dimensional periodic structure. Accordingly, the idea is to interpret the studied PDE as the “restriction” of an augmented PDE in higher dimensions, where periodicity along the interface is recovered. This so-called lifting approach allows one to extend the previously developed ideas, but comes with the price that the augmented equation is non-elliptic (in the sense of the principal part of the differential operator), and thus more complicated to analyse and to solve numerically. Numerical results will be provided toillustrate the method.