On the Minimal Shift in the Shifted Laplacian Preconditioner for Multigrid to Work
Over the past years, the shifted Laplacian has been advocated as a way of making multigrid work for the indefinite Helmholtz equation. The idea is to use a shift into the complex plane of the wave number in the operator, and then to use the shifted operator as a preconditioner for a Krylov method. 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. There are however two conflicting requirements here: 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 rigorously proved last year that the preconditioner is good if the shift is at most of the size of the wavenumber. We prove here rigorously that if the shift is less than the size of the wavenumber squared, multigrid will not work. It is therefore not possible to solve the shifted Laplace preconditioner with multigrid in the regime where it is a good preconditioner.
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