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

We analyse two-level Schwarz domain-decomposition GMRES preconditioners – both the classic additive Schwarz preconditioner and a hybrid variant – 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 – up to 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 bounded independently of $k$. These are the first convergence results about a two-level Schwarz preconditioner for high-frequency Helmholtz with a coarse space that is pollution free (up to factors of $\log k$) and does not consist of problem-adapted basis functions. Additionally, these are the first $k$-explicit convergence results about any two-level completely-additive Schwarz preconditioner for high-frequency Helmholtz.