Fourier--Galerkin Methods for Subwavelength Resonances in 2D Acoustic Metamaterials
This work provides an efficient computational framework for resonance problems in acoustic metamaterials, which is relevant for researchers in wave propagation and metamaterial design.
The authors develop a Fourier-Galerkin method that reduces the computation of subwavelength resonances in 2D acoustic metamaterials to a low-dimensional nonlinear eigenvalue problem, avoiding large-scale discretizations. The method yields explicit asymptotic expansions and uses FFT-based quadrature for fast evaluation.
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 $ω$ 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.