On the minimal shift in the shifted Laplacian preconditioner for multigrid to work

with the same efficiency, but it turned out that this is a very difficult task. Textbooks mention that there are substantial difficulties, see [3, page 72], [11, page 212], [12, page 400], and also the review [7] for why in general iterative methods have difficulties when applied to the Helmholtz equation (1). Motivated by the early proposition in [2] to use the Laplacian to precondition the Helmholtz equation, the shifted Laplacian has been advocated over the past decade as a way of making multigrid work for the indefinite Helmholtz equation, see [6, 10, 1, 5, 4] and references therein. The idea is to shift the wave number into the complex plane to obtain a good preconditioner for a Krylov method when solving (1). The hope is that due to the shift, it becomes possible to use standard multigrid to invert the preconditioner, and if the shift is not too big, it is still an effective preconditioner for the Helmholtz equation with a real wave number. This implies however two conflicting requirements: the shift should be not too large for the shifted preconditioner to be a good preconditioner, and it should be large enough for multigrid to work. It was already indicated in [7] that it is not possible to satisfy both these requirements, see also [4]. It was then rigorously proved in [9] that the preconditioner is effective, i.e. iteration numbers stay bounded independently of the wave number k if the shift is at most of the size of the wavenumber. We prove here rigorously for a one dimensional model problem that if the complex shift is less than the size of the wavenumber squared, multigrid will not work. It is therefore not pos-