2025/05/22 by Victorita Dolean, Pierre Marchand, Dolean, Victorita +5 · 1 citation
Engineering · Mathematics · Physics and Astronomy · #65F10 (Primary) #65N22 #65N30 (Secondary) #Electromagnetic Scattering and Analysis #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2505.16345
openalex publication_date 2025/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The finite element solution of Helmholtz problems near resonant or quasi-resonant frequencies poses significant challenges, as iterative solvers typically suffer from severely degraded convergence. We analyze the convergence behavior of GMRES applied to linear systems arising from such configurations. Theoretical convergence estimates are derived based on harmonic Ritz values, highlighting their proximity to small eigenvalues as a key determining factor. We further examine deflation strategies and their interplay with preconditioning techniques, using the Complex Shifted Laplacian preconditioner as a case study. Numerical experiments on resonant and quasi-resonant test cases validate the theoretical framework and demonstrate the effectiveness of deflation strategies. This study provides new insights and practical guidance for analyzing and improving iterative solvers for time-harmonic problems near resonances.