École Nationale Supérieure de
Techniques Avancées

Abstract

The interest in quasiperiodic media has increased since the recent discovery of quasicrystals. The mathematical theory of quasiperiodic functions is well-known. For example, a 1D quasiperiodic function of order n is seen as the trace of a periodic function of n variables along a given direction. PDEs with quasiperiodic coefficients have also been deeply investigated. However in order to numerically solve these PDEs, little work seems to have been done aside from the homogenization setting. Our goal is to extend the previous works of Gérard-Varet and Masmoudi (2012) as well as Blanc, Le Bris and Lions (2015) to boundary value problems to which homogenization theory cannot be applied. We consider the one-dimensional Helmholtz boundary value problem with quasiperiodic coefficients, in the case where these coefficients can be seen as the trace of a 2D function along an irrational direction. We show that one can equivalently solve a 2D non-elliptic PDE with periodic coefficients. This PDE is numerically solved using the method developed by Fliss and Joly (2009), and we identify the influence of different parameters through simulations.