Local Fourier Analysis of the Complex Shifted Laplacian preconditioner for Helmholtz problems

TitleLocal Fourier Analysis of the Complex Shifted Laplacian preconditioner for Helmholtz problems
Publication TypeJournal Article
Year of PublicationIn Press
AuthorsCools S, Vanroose W
Refereed DesignationUnknown
JournalNumerical Linear Algebra with Applications
KeywordsHelmholtz equation, high wavenumber, indefinite systems, Krylov method, Local Fourier Analysis, Shited Laplacian preconditioner
Abstract

The class of Shifted Laplacian preconditioners are known to significantly speed-up Krylov convergence.
However, these preconditioners have a parameter beta, a measure of the complex shift. Due to
contradictory requirements for the multigrid and Krylov convergence, the choice of this shift parameter
can be a bottleneck in applying the method. In this paper, we propose a wavenumber-dependent minimal
complex shift parameter which is predicted by a rigorous k-grid Local Fourier Analysis (LFA) of the
multigrid scheme. We claim that, given any (regionally constant) wavenumber, this minimal complex shift
parameter provides the reader with a parameter choice that leads to efficient Krylov convergence. Numerical
experiments in one and two spatial dimensions validate the theoretical results. It appears that the proposed
complex shift is both the minimal requirement for a multigrid V-cycle to converge, as well as being nearoptimal
in terms of Krylov iteration count.

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