Fourier--Galerkin Methods for Subwavelength Resonances in two-dimensional Acoustic Metamaterials

Abstract

We present a Fourier--Galerkin asymptotic framework for the analysis and computation of subwavelength resonances in two-dimensional scattering problems in finite domains. Starting from the boundary integral formulation, we apply a Fourier--Galerkin discretization to derive an explicit finite-dimensional effective matrix whose kernel characterizes the resonant frequencies. In the subwavelength regime, we obtain asymptotic expansions of this matrix in terms of $ω$ and the material contrast, identifying the leading-order operators and their kernel structure. This reduction transforms the resonance problem into a low-dimensional nonlinear eigenvalue problem, avoiding large-scale discretizations and global root-search procedures. The entries of the effective matrix are explicitly computable and admit fast evaluation using FFT-based quadrature. The resulting approach provides an efficient and robust computational framework for resonances in general smooth geometries.