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

· 2025 · math.NA · arXiv 2501.15976

2 Pith papers cite this work. Polarity classification is still indexing.

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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.

fields

math.NA 2

years

2026 2

verdicts

UNVERDICTED 2

representative citing papers

Sobolev stability of the $L^2$-projection on hybrid meshes

math.NA · 2026-07-02 · unverdicted · novelty 7.0

Establishes L^p- and W^{1,p}-stability of the L2-projection on hybrid meshes for all K >= 2 in Q-RG and Q-RB refinements, extending prior results limited to parallelograms and K <= 9.

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Showing 2 of 2 citing papers.

  • Sobolev stability of the $L^2$-projection on hybrid meshes math.NA · 2026-07-02 · unverdicted · none · ref 26 · internal anchor

    Establishes L^p- and W^{1,p}-stability of the L2-projection on hybrid meshes for all K >= 2 in Q-RG and Q-RB refinements, extending prior results limited to parallelograms and K <= 9.

  • Spectral coarse spaces based on indefinite operators: the $H_k$-GenEO method math.NA · 2026-05-29 · unverdicted · none · ref 19 · internal anchor

    H_k-GenEO constructs spectral coarse spaces from indefinite local eigenproblems to precondition highly indefinite PDEs, providing sufficient conditions for GMRES robustness and observed practical stability as k grows.