Two-level hybrid Schwarz preconditioners with piecewise-polynomial coarse spaces for the high-frequency Helmholtz equation

Abstract: We analyse two-level hybrid Schwarz domain-decomposition GMRES preconditioners for finite-element discretisations of the Helmholtz equation with wavenumber $k$, where the coarse space consists of piecewise polynomials. We prove results for fixed polynomial degree (in both the fine and coarse spaces), as well as for polynomial degree increasing like $\log k$. In the latter case, we exhibit choices of fine and coarse spaces such that, modulo factors of $\log k$, the fine and coarse spaces are both 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 constant; i.e., modulo the important question of how to efficiently solve the coarse problem, this is the (arguably) theoretically ideal situation. Along with the results in the companion paper Galkowski, Spence, arXiv:2501.11060, these are the first rigorous convergence results about a two-level Schwarz preconditioner applied to the high-frequency Helmholtz equation with a coarse space that does not consist of problem-adapted basis functions.