Title
Local Fourier analysis of the complex shifted Laplacian preconditioner for Helmholtz problems
Author
Faculty/Department
Faculty of Sciences. Mathematics and Computer Science
Publication type
article
Publication
Chichester :Wiley ,
Subject
Mathematics
Source (journal)
Numerical linear algebra with applications. - Chichester
Volume/pages
20(2013) :4 , p. 575-597
ISSN
1070-5325
ISI
000322031700004
Carrier
E
Target language
English (eng)
Full text (Publishers DOI)
Affiliation
University of Antwerp
Abstract
In this paper, we solve the Helmholtz equation with multigrid preconditioned Krylov subspace methods. The class of shifted Laplacian preconditioners is known to significantly speed up Krylov convergence. However, these preconditioners have a parameter inline image, a measure of the complex shift. Because of 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 and being near optimal in terms of Krylov iteration count.
Full text (open access)
https://repository.uantwerpen.be/docman/irua/42703d/e284ee35.pdf
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