A nonuniform mesh method in the Floquet parameter domain for wave scattering by periodic surfaces

In this paper, we propose a new numerical method to simulate acoustic scattering problems in two-dimensional periodic structures with non-periodic incident fields. Applying the Floquet-Bloch transform to the scattering problem yields a family of quasi-periodic boundary value problems dependent on the Floquet-Bloch parameter. Consequently, the solution of the original scattering problem is written as the inverse Floquet-Bloch transform of the solutions to these boundary value problems. The key step in our method is the numerical approximation of this integral transform by a quadrature rule with a nonuniform choice of quadrature points adapted to the regularity of the family of quasi-periodic solutions. This achieved by a graded subdivision of the full interval for the Floquet-Bloch parameter and applying a Gauss-Legrendre quadrature rule on each subinterval. We prove that the numerical method converges exponentially with respect to both the number of subintervals and the number of Gaussian quadrature points. Some numerical experiments are provided to illustrate the results.