Scattering problems in periodic or locally perturbed periodic domains: Convergence of the perfectly matched layer approximation and efficient numerical solution

We study scattering of a time-harmonic acoustic wave by a locally perturbed periodic surface, where the incident field need not be periodic. This scattering problem is challenging due to the unboundedness of the domain and the lack of periodicity. Using a coordinate transformation, we remove the perturbation and obtain an equation with non-constant coefficients in a periodic domain. This allows application of the Floquet–Bloch transform, reducing the problem to a coupled family (indexed by the Floquet parameter) of periodic problems in a bounded cell. We approximate the solution with the perfectly matched layer (PML) method and prove the exponential convergence of this approximation on compact sets for an incident spherical wave. Moreover, we show that the Floquet–Bloch transform of the solution of the PML problem is analytic with respect to the Floquet parameter. This enables us to accurately discretize the inverse Floquet–Bloch transform using only a few values of the Floquet parameter. Finally, by exploiting the system structure via the Schur complement, we propose a fast parallel solver and illustrate its efficiency by numerical examples.