Non-periodic acoustic and electromagnetic, scattering from periodic structures in 3D

Scattering of non-periodic waves from unbounded structures is difficult to treat, as one typically formulates the problem in an unbounded domain covering the unbounded periodic structure. The Floquet-Bloch transform reduces the latter problem to a family of decoupled periodic scattering problems. This reduction is in particular interesting from the point of view of numerical computations. We analyze a corresponding fully discrete numerical solution algorithm for three-dimensional scattering problems in acoustics and electromagnetics, proving in particular convergence rates under suitable assumptions on the geometry and the material coefficients. A crucial part of our analysis relies on the continuous dependence of the family of solutions to the quasiperiodic scattering problems on the quasiperiodicity. The latter part is actually more difficult to establish than for corresponding two-dimensional problems. We further provide a numerical example in 3D acoustics that illustrates feasibility of the proposed algorithm.