We present a fast spectral Galerkin scheme for the discretization of boundary integral equations arising from two-dimensional Helmholtz transmission problems in multi-layered periodic structures or gratings. Employing suitably parametrized Fourier basis and excluding Rayleigh-Wood anomalies, we rigorously establish the well-posedness of both continuous and discrete problems, and prove super-algebraic error convergence rates for the proposed scheme. Through several numerical examples, we confirm our findings and show performances competitive to those attained via Nystr\"om methods.

中文翻译：

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分层介质中准周期性二维亥姆霍兹散射的快速求解器

我们提出了一种快速谱伽辽金方案，用于离散化由多层周期结构或光栅中的二维亥姆霍兹传输问题引起的边界积分方程。采用适当的参数化傅里叶基并排除瑞利伍德异常，我们严格地建立了连续和离散问题的适定性，并证明了所提出方案的超代数误差收敛率。通过几个数值示例，我们证实了我们的发现，并展示了与通过 Nystr\"om 方法获得的性能相比具有竞争力的性能。