Fourier--Galerkin Methods for Subwavelength Resonances in 2D Acoustic Metamaterials

We present a Fourier--Galerkin framework for the analysis and computation of subwavelength resonances in two-dimensional scattering problems in finite domains. Starting from the boundary integral formulation, we project the operator onto Fourier modes and derive an explicit finite-dimensional effective matrix whose singularity characterizes the resonant frequencies. In the subwavelength regime, we obtain asymptotic expansions of this matrix in terms of $\omega$ 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.