Development and frequency-explicit theory of two-level Helmholtz domain-decomposition methods

Develop and establish a frequency-explicit theory for two-level domain-decomposition methods for high-frequency wave problems, including the high-frequency Helmholtz equation, while advancing the development of effective two-level methods for such problems.

Background

The paper studies two-level hybrid Schwarz domain-decomposition preconditioners for finite-element discretizations of the high-frequency Helmholtz equation. Existing analyses generally assume that the coarse Galerkin space lies in the asymptotic regime, where the Galerkin solution is uniformly quasi-optimal with respect to the wavenumber. The paper instead proves convergence results when the piecewise-polynomial coarse space is only preasymptotic, under a wavenumber-dependent mesh condition.

The authors frame the broader development of two-level domain-decomposition methods and their frequency-explicit analysis for high-frequency wave problems as unresolved. The results address one part of this broader problem by establishing logarithmic iteration bounds for a specific class of hybrid Schwarz preconditioners and preasymptotic coarse spaces, but do not resolve the general development and theory of such methods.

References

However, both the development of and the frequency-explicit theory of two-level DD methods for high-frequency wave problems, such as the high-frequency Helmholtz equation, are longstanding open problems (see, e.g., the recent computational studies and the references therein).

— Two-level hybrid Schwarz preconditioners with preasymptotic piecewise-polynomial coarse spaces for the high-frequency Helmholtz equation  (2610.11774 - Galkowski et al., 8 Oct 2026) in Section 1, “Context and informal statement of the main result”