Wave propagation in quasiperiodic media

Abstract

A quasiperiodic medium is an ordered medium without being periodic. A fairly well-known example since the 2011 Nobel Prize in Chemistry is the quasicrystal. The notion of quasi-periodic and more generally almost periodic function is a very well-defined notion in the mathematical literature. To give an idea, a 1D quasiperiodic function is the trace along a line of a periodic function of n variables. PDEs with quasi-periodic coefficients have been the subject of a number of theoretical studies, but it seems that there has been much less work on the numerical resolution of these equations.The objective of this work is to develop original numerical methods for the resolution of the time-harmonic wave equation in quasiperiodic media, in the spirit of the methods previously developed for periodic media. The idea is to use the fact that the study of an elliptic PDE with quasiperiodic coefficients comes down to the study of an "augmented" non-elliptic PDE in higher dimensions, but whose coefficients are periodic. This so-called lifting approach allows to use tools that are adapted for periodic media, but comes with the price that the augmented PDE is non-elliptic, in the sense of its principal part.In this talk, I will first present the lifting method for the 1D Helmholtz equation with dissipation. I will then explain how this method can be used to solve the 2D Helmholtz equation in a junction of periodic media cut in an arbitrary direction. The method will be illustrated by numerical results.