How Large a Shift is Needed in the Shifted Helmholtz Preconditioner for its Effective Inversion by Multigrid?

The shifted Helmholtz operator has received a lot of attention over the past decade as a preconditioner for the iterative solution of the Helmholtz equation. The idea is that if one uses a small complex shift, the shifted Helmholtz operator is still close to the original Helmholtz operator and could thus be an effective preconditioner. It was shown in [M. J. Gander, I. G. Graham, and E. A. Spence, Numer. Math., 53 (2015), pp. 573--579] that the shift can be at most $O(k)$ to prove rigorously wave number independent convergence of the preconditioned system solved with GMRES, provided the preconditioner is inverted exactly. In practice, however, the preconditioner is inverted only approximately, and if one shifts enough, this can be done effectively by standard multigrid methods. We show in this paper that for a finite element discretization, the shift has to be at least O(k²) to be able to invert the shifted Helmholtz preconditioner using multigrid. There is therefore a gap between being a good preconditioning operator (shift at most $O(k)$) and being able to effectively invert the preconditioner by multigrid (shift at least O(k²)). So what shift should be chosen in practice, and when is the preconditioner not inverted exactly? By studying the numerical range of the preconditioned operator, we show that one cannot obtain analytical results for this case with currently available tools. We thus test the preconditioner extensively numerically for a wave guide type square domain in the range of shifts between O(√k) and O(k²) with approximate inversion by one multigrid V-cycle. We find in our experiments that preconditioned GMRES iteration numbers will then inevitably grow like O(k²). We also see that in contrast to common practice where shifts of O(k²) are used, it might be beneficial for the wave guide to use a smaller shift, e.g., O(k3/2), especially when several smoothing steps are used.