Theory of two-level Schwarz preconditioners with piecewise-polynomial coarse spaces for the high-frequency Helmholtz equation

Mathematics > Numerical Analysis [Submitted on 27 Jan 2025 (v1), last revised 18 Jun 2026 (this version, v4)] Title:Theory of two-level Schwarz preconditioners with piecewise-polynomial coarse spaces for the high-frequency Helmholtz equation View PDF HTML (experimental)Abstract:We analyse the classic two-level additive Schwarz domain-decomposition GMRES preconditioner for finite-element discretisations of the Helmholtz equation with large wavenumber $k$, where both the fine and coarse spaces consist of piecewise polynomials with polynomial degree increasing like $\log k$. We exhibit choices of these fine and coarse spaces such that -- up to factors of $\log k$ -- both are pollution free (with the ratio of the coarse-space dimension to the fine-space dimension arbitrarily small), the number of degrees of freedom per subdomain is constant, and the number of GMRES iterations is proved to be bounded independently of $k$. These are the first $k$-explicit convergence results about a two-level Schwarz preconditioner for high-frequency Helmholtz with a coarse space that is pollution free and does not consist of problem-adapted basis functions. Submission history From: Euan Spence [view email][v1] Mon, 27 Jan 2025 11:49:59 UTC (46 KB) [v2] Wed, 19 Mar 2025 18:48:25 UTC (45 KB) [v3] Fri, 26 Sep 2025 16:04:57 UTC (54 KB) [v4] Thu, 18 Jun 2026 10:00:51 UTC (55 KB) Current browse context: math.NA References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.