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

TL;DR

This paper develops a wavenumber-dependent minimal complex shift parameter for the Shifted Laplacian preconditioner in Helmholtz problems, using Local Fourier Analysis to optimize multigrid and Krylov convergence.

Contribution

It introduces a rigorous, wavenumber-specific shift parameter prediction method based on Local Fourier Analysis, improving the efficiency of Helmholtz solvers.

Findings

The proposed shift parameter ensures multigrid convergence.

Numerical results validate the near-optimality of the shift in Krylov iterations.

The method applies to both 1D and 2D Helmholtz problems.

Abstract

In this paper we solve the Helmholtz equation with multigrid preconditioned Krylov subspace methods. 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 near-optimal in terms of Krylov iteration count.