Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: what is the largest shift for which wavenumber-independent convergence is guaranteed?

There has been much recent research on preconditioning discretisations of the Helmholtz operator ∆+ k (subject to suitable boundary conditions) using a discrete version of the so-called “shifted Laplacian” ∆+(k+iε) for some ε > 0. This is motivated by the fact that, as ε increases, the shifted problem becomes easier to solve iteratively. Despite many numerical investigations, there has been no rigorous analysis of how to chose the shift. In this paper, we focus on the question of how large ε can be so that the shifted problem provides a preconditioner that leads to k-independent convergence of GMRES, and our main result is a sufficient condition on ε for this property to hold. This result holds for finite element discretisations of both the interior impedance problem and the sound-soft scattering problem (with the radiation condition in the latter problem imposed as a far-field impedance boundary condition). Note that we do not address the important question of how large ε should be so that the preconditioner can easily be inverted by standard iterative methods.