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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| Main Authors: | , , , |
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| Format: | Article (Journal) |
| Language: | English |
| Published: |
22 November 2025
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| In: |
Journal of scientific computing
Year: 2025, Volume: 105, Issue: 3, Pages: 1-28 |
| ISSN: | 1573-7691 |
| DOI: | 10.1007/s10915-025-03138-y |
| Online Access: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.1007/s10915-025-03138-y
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| Author Notes: | Chupeng Ma, Christian Alber, Robert Scheichl, Yongwei Zhang |
| Summary: | 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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| Item Description: | Online veröffentlicht: 22. November 2025 Gesehen am 27.01.2026 |
| Physical Description: | Online Resource |
| ISSN: | 1573-7691 |
| DOI: | 10.1007/s10915-025-03138-y |