Two-level restricted additive Schwarz preconditioner based on multiscale spectral generalized FEM for heterogeneous Helmholtz problems
We present and analyze a two-level restricted additive Schwarz (RAS) preconditioner for heterogeneous Helmholtz problems, based on a multiscale spectral generalized finite element method (MS-GFEM) proposed in [C. Ma, C. Alber, and R. Scheichl, SIAM. J. Numer. Anal., 61 (2023), pp. 1546-1584]. The pr...
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| Hauptverfasser: | , , , |
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| Dokumenttyp: | Article (Journal) |
| Sprache: | Englisch |
| Veröffentlicht: |
22 November 2025
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| In: |
Journal of scientific computing
Year: 2025, Jahrgang: 105, Heft: 3, Pages: 1-28 |
| ISSN: | 1573-7691 |
| DOI: | 10.1007/s10915-025-03138-y |
| Online-Zugang: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.1007/s10915-025-03138-y
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| Verfasserangaben: | Chupeng Ma, Christian Alber, Robert Scheichl, Yongwei Zhang |
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| 245 | 1 | 0 | |a Two-level restricted additive Schwarz preconditioner based on multiscale spectral generalized FEM for heterogeneous Helmholtz problems |c Chupeng Ma, Christian Alber, Robert Scheichl, Yongwei Zhang |
| 264 | 1 | |c 22 November 2025 | |
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| 520 | |a We present and analyze a two-level restricted additive Schwarz (RAS) preconditioner for heterogeneous Helmholtz problems, based on a multiscale spectral generalized finite element method (MS-GFEM) proposed in [C. Ma, C. Alber, and R. Scheichl, SIAM. J. Numer. Anal., 61 (2023), pp. 1546-1584]. The preconditioner uses local solves with impedance boundary conditions, and a global coarse solve based on the MS-GFEM approximation space constructed from local eigenproblems. It is derived by first formulating MS-GFEM as a Richardson iterative method, and without using an oversampling technique, reduces to the preconditioner recently proposed and analyzed in [Q. Hu and Z.Li, arXiv 2402.06905]. We prove that both the Richardson iterative method and the preconditioner used within GMRES converge at a rate of $$\Lambda $$under some reasonable conditions, where $$\Lambda $$denotes the error of the underlying MS-GFEM approximation. Notably, the convergence proof of GMRES does not rely on the ‘Elman theory’. An exponential convergence property of MS-GFEM, resulting from oversampling, ensures that only a few iterations are needed for convergence with a small coarse space. In particular, in the constant-coefficient, non-trapping case, with $$h\sim k^{-1-\gamma }$$for some $$\gamma \in (0,1]$$, it holds that $$\Lambda \sim k^{-1+\frac{\gamma }{2}}$$, with the coarse-space dimension $$\sim k^{d}\log ^{d}(k)$$. We present extensive numerical experiments to illustrate the performance of the preconditioner, including 2D and 3D benchmark geophysics tests, and a high-contrast coefficient example arising in applications. | ||
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| 650 | 4 | |a Domain decomposition | |
| 650 | 4 | |a Helmholtz equation | |
| 650 | 4 | |a Restricted additive Schwarz | |
| 650 | 4 | |a Spectral coarse space | |
| 650 | 4 | |a Two-level method | |
| 700 | 1 | |a Alber, Christian |e VerfasserIn |0 (DE-588)1303968878 |0 (DE-627)1860278698 |4 aut | |
| 700 | 1 | |a Scheichl, Robert |d 1972- |e VerfasserIn |0 (DE-588)1173753842 |0 (DE-627)1043602305 |0 (DE-576)515668532 |4 aut | |
| 700 | 1 | |a Zhang, Yongwei |e VerfasserIn |4 aut | |
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