Convergence analysis of GMRES applied to Helmholtz problems near resonances
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915297927102464 |
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| author | Dolean, Victorita Marchand, Pierre Modave, Axel Raynaud, Timothée |
| author_facet | Dolean, Victorita Marchand, Pierre Modave, Axel Raynaud, Timothée |
| contents | In this work we study how the convergence rate of GMRES is influenced by the properties of linear systems arising from Helmholtz problems near resonances or quasi-resonances. We extend an existing convergence bound to demonstrate that the approximation of small eigenvalues by harmonic Ritz values plays a key role in convergence behavior. Next, we analyze the impact of deflation using carefully selected vectors and combine this with a Complex Shifted Laplacian preconditioner. Finally, we apply these tools to two numerical examples near (quasi-)resonant frequencies, using them to explain how the convergence rate evolves. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_16345 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convergence analysis of GMRES applied to Helmholtz problems near resonances Dolean, Victorita Marchand, Pierre Modave, Axel Raynaud, Timothée Numerical Analysis 65F10 (Primary), 65N22, 65N30 (Secondary) In this work we study how the convergence rate of GMRES is influenced by the properties of linear systems arising from Helmholtz problems near resonances or quasi-resonances. We extend an existing convergence bound to demonstrate that the approximation of small eigenvalues by harmonic Ritz values plays a key role in convergence behavior. Next, we analyze the impact of deflation using carefully selected vectors and combine this with a Complex Shifted Laplacian preconditioner. Finally, we apply these tools to two numerical examples near (quasi-)resonant frequencies, using them to explain how the convergence rate evolves. |
| title | Convergence analysis of GMRES applied to Helmholtz problems near resonances |
| topic | Numerical Analysis 65F10 (Primary), 65N22, 65N30 (Secondary) |
| url | https://arxiv.org/abs/2505.16345 |