Local Fourier analysis of the complex shifted Laplacian preconditioner for Helmholtz problems

Abstract

SUMMARYIn 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 β MathClass-rel∈ double-struckR, 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. Copyright © 2013 John Wiley & Sons, Ltd.

An improved two‐grid preconditioner for the solution of three‐dimensional Helmholtz problems in heterogeneous media

“…Finally, it has been shown that reasonably good smoothing factors for the Jacobi smoother can be obtained on all the grid hierarchy for the complex shifted Laplacian operator in three dimensions; see also , where a local Fourier analysis of the damped Jacobi method is performed on the complex shifted Laplacian in one and two dimensions. With the selected relaxation parameters, we now investigate the spectrum of preconditioned Helmholtz matrices.…”

Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: what is the largest shift for which wavenumber-independent convergence is guaranteed?

“…(Using the notation above, preconditioning with the second operator corresponds to choosing ε ∼ k 2 and constructing B −1 ε using a multigrid V-cycle.) Preconditioning with ( + k 2 + iε) −1 and ε ∼ k 2 was then further investigated in the context of multigrid in [8] and [49].…”

Multigrid methods for Helmholtz problems: A convergent scheme in 1D using standard components

“…Proceeding in a different direction, Elman, Ernst and O'Leary studied the convergence problems of the classical multigrid algorithm applied to the Helmholtz equation in more detail by spectral analysis in [17], and used Krylov methods, both as smoothers and as outer convergence accelerator for multigrid, in order to obtain a practical and more robust method. One can also use the Laplacian, as proposed by Bayliss, Goldstein and Turkel [5], or the shifted Laplacian, as advocated by Erlangga, Vuik and Oosterlee [21], as a preconditioner for the Helmholtz equation, and then apply the preconditioner using multigrid, see also [32], and for an algebraic version [2], where it is shown numerically that, asymptotically, the number of iterations grows linearly with the wave number for this approach, see also [15,48]. This illustrates well that in the shifted Laplacian preconditioner, there are two conflicting requirements: the shift should be large enough in order for multigrid to work, and not too large, in order to still have a good preconditioner of the original problem, and one cannot, in general, satisfy both; see [22] for a simple Fourier analysis argument, and [31] for a complete theoretical treatment, and the preprint [15] for a detailed local Fourier analysis and a thorough numerical study.…”