This work, done in collaboration with Pierre Amenoagbadji (Columbia University) and Patrick Joly (POEMS), is devoted to the solution of the Helmholtz equation in 1D unbounded quasiperiodic media. By this we mean that the coefficients appearing in the model are quasiperiodic functions of the 1D space variable, namely the trace along a line of a periodic function of n variables.

When the coefficients are periodic (which is a special case), several methods have been proposed to characterize and compute the solution. However, when the coefficients are quasi-periodic without being periodic, the above methods cannot be applied directly.

We use an original method, that we call the lifting method, which has been used in several papers on homogenization theory. The original problem can thus be lifted to an nD « augmented » problem with periodic coefficients, and the 1D solution is the trace along this line of the nD solution. The advantage is that the periodicity of the augmented problem enables to use the ideas proposed for solving periodic Helmholtz equations in periodic waveguides. However, since the augmented equation is a degenerate elliptic equation, the corresponding tools have to be adapted since new difficulties arise both in the analysis and in the design of the resulting numerical method.

I will present our results for the Helmholtz equation with dissipation (where the solution decays at infinity) and then for the equation without dissipation (where the solution can propagate to infinity), analyzing the latter case using a limiting absorption principle.