A Floquet-Bloch transform based numerical method for scattering from locally perturbed periodic surfaces

TL;DR

This paper introduces a novel numerical method based on the Floquet-Bloch transform for simulating wave scattering from locally perturbed periodic surfaces, offering convergence analysis and practical implementation details.

Contribution

The paper develops a new Floquet-Bloch transform-based numerical scheme for scattering from locally perturbed periodic surfaces, including convergence analysis and efficient implementation strategies.

Findings

Convergence and error bounds established for the Galerkin discretization.

The method effectively handles locally perturbed periodic structures.

Numerical examples demonstrate the scheme's accuracy and efficiency.

Abstract

Scattering problems for periodic structures have been studied a lot in the past few years. A main idea for numerical solution methods is to reduce such problems to one periodicity cell. In contrast to periodic settings, scattering from locally perturbed periodic surfaces is way more challenging. In this paper, we introduce and analyze a new numerical method to simulate scattering from locally perturbed periodic structures based on the Bloch transform. As this transform is applied only in periodic domains, we firstly rewrite the scattering problem artificially in a periodic domain. With the help of the Bloch transform, we secondly transform this problem into a coupled family of quasiperiodic problems posed in the periodicity cell. A numerical scheme then approximates the family of quasiperiodic solutions (we rely on the finite element method) and backtransformation provides the solution to the original scattering problem. In this paper, we give convergence analysis and error bounds for a Galerkin discretization in the spatial and the quasiperiodicity's unit cells. We also provide a simple and efficient way for implementation that does not require numerical integration in the quasiperiodicity, together with numerical examples for scattering from locally perturbed periodic surfaces computed by this scheme.