Can Symmetric Positive Definite (SPD) coarse spaces perform well for indefinite Helmholtz problems?

[2403.18378] Can Symmetric Positive Definite (SPD) coarse spaces perform well for indefinite Helmholtz problems? Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Numerical Analysis arXiv:2403.18378 (math) [Submitted on 27 Mar 2024 ( v1 ), last revised 6 Feb 2026 (this version, v5)] Title: Can Symmetric Positive Definite (SPD) coarse spaces perform well for indefinite Helmholtz problems? Authors: Victorita Dolean , Mark Fry , Matthias Langer View a PDF of the paper titled Can Symmetric Positive Definite (SPD) coarse spaces perform well for indefinite Helmholtz problems?, by Victorita Dolean and 2 other authors View PDF HTML (experimental) Abstract: Wave propagation problems governed by the Helmholtz equation remain among the most challenging in scientific computing, due to their indefinite nature. Domain decomposition methods with spectral coarse spaces have emerged as some of the most effective preconditioners, yet their theoretical guarantees often lag behind practical performance. In this work, we introduce and analyse the $\Delta_k$-GenEO coarse space within the two-level additive Schwarz preconditioners for heterogeneous Helmholtz problems. This is an adaptation of the $\Delta$-GenEO coarse space. Our results sharpen the $k$-explicit conditions for GMRES convergence, reducing the restrictions on the subdomain size and eigenvalue threshold. This narrows the long-standing gap between pessimistic theory and empirical evidence, and reveals why GenEO spaces based on SPD (symmetric positive definite) eigenvalue problems remain surprisingly effective despite their apparent limitations. Numerical experiments confirm the theory, demonstrating scalability, robustness to heterogeneity for low to moderate frequencies (while experiencing limitations in the high frequency cases), and significantly milder coarse-space growth than conservative estimates predict. Subjects: Numerical Analysis (math.NA) MSC classes: 65N22, 65N55, 65F10 Cite as: arXiv:2403.18378 [math.NA] (or arXiv:2403.18378v5 [math.NA] for this version) https://doi.org/10.48550/arXiv.2403.18378 Focus to learn more arXiv-issued DOI via DataCite Journal reference: J. Comput. Appl. Math. 484 (2026), 117403 Related DOI : https://doi.org/10.1016/j.cam.2026.117403 Focus to learn more DOI(s) linking to related resources Submission history From: Victorita Dolean [ view email ] [v1] Wed, 27 Mar 2024 09:17:38 UTC (15 KB) [v2] Tue, 18 Jun 2024 08:41:37 UTC (15 KB) [v3] Thu, 30 Oct 2025 12:26:54 UTC (59 KB) [v4] Sat, 1 Nov 2025 12:33:31 UTC (45 KB) [v5] Fri, 6 Feb 2026 06:11:32 UTC (65 KB) Full-text links: Access Paper: View a PDF of the paper titled Can Symmetric Positive Definite (SPD) coarse spaces perform well for indefinite Helmholtz problems?, by Victorita Dolean and 2 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.NA < prev | next > new | recent | 2024-03 Change to browse by: cs cs.NA math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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