A two-level shifted Laplace preconditioner for Helmholtz problems: Field-of-values analysis and wavenumber-independent convergence

One of the main tools for solving linear systems arising from the discretization of the Helmholtz equation is the shifted Laplace preconditioner, which results from the discretization of a perturbed Helmholtz problem $-\Delta u - (k^2 + i \varepsilon )u = f$ where $0 \neq \varepsilon \in \mathbb{R}$ is an absorption parameter. In this work we revisit the idea of combining the shifted Laplace preconditioner with two-level deflation and apply it to Helmholtz problems discretized with linear finite elements. We use the convergence theory of GMRES based on the field of values to prove that GMRES applied to the two-level preconditioned system with a shift parameter $\varepsilon \sim k^2$ converges in a number of iterations independent of the wavenumber $k$,provided that the coarse mesh size $H$ satisfies a condition of the form $Hk^{2} \leq C$ for some constant $C$ depending on the domain but independent of the wavenumber $k$. This behaviour is sharply different to the standalone shifted Laplacian, for which wavenumber-independent GMRES convergence has been established only under the condition that $\varepsilon \sim k$ by [M.J. Gander, I.G. Graham and E.A. Spence, Numer. Math., 131 (2015), 567-614]. Finally, we present numerical evidence that wavenumber-independent convergence of GMRES also holds for pollution-free meshes, where the coarse mesh size satisfies $Hk^{3/2} \leq C $, and inexact coarse grid solves.