How should one choose the shift for the shifted Laplacian to be a good preconditioner for the Helmholtz equation?

There has been much recent research on preconditioning discretisations of the Helmholtz operator ∆+ k (subject to suitable boundary conditions) with the inverse of a discrete version of the so-called “shifted Laplacian” ∆+ (k + iε) for some ε > 0. (In practice this inverse is replaced with a cheaper approximation in order to obtain a practically viable preconditioner.) Despite many numerical investigations, there has been no rigorous analysis of how one should chose the shift. In this paper we give sufficient conditions on ε for the matrix of the shifted problem to be a good preconditioner for the original matrix as k → ∞. The results hold 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).