Analyzing the wave number dependency of the convergence rate of a multigrid preconditioned Krylov method for the Helmholtz equation with an absorbing layer

Abstract

SUMMARY This paper analyzes the Krylov convergence rate of a Helmholtz problem preconditioned with multigrid. The multigrid method is applied to the Helmholtz problem formulated on a complex contour and uses the generalized minimal residual method as a smoother substitute at each level. A one‐dimensional model is analyzed both in a continuous and discrete way. It is shown that the Krylov convergence rate of the continuous problem is independent of the wave number. The discrete problem, however, can deviate significantly from this bound because of a pitchfork in the spectrum. It is further shown in numerical experiments that the convergence rate of the Krylov method approaches the continuous bound as the grid distance h gets small. Copyright © 2012 John Wiley & Sons, Ltd.

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

“…Upon reaching this maximum, the number of iterations decreases slowly, exhibiting a steep descend for values of σ around − 8 ∕ h 2 . The same observation has been reported in , where an eigenvalue analysis was performed to rigourously anticipate the Krylov convergence behaviour. Note that we have used standard Dirichlet boundary conditions, as opposed to the absorbing boundary conditions used in ; however, the conclusions are identical.…”

“…The same observation has been reported in , where an eigenvalue analysis was performed to rigourously anticipate the Krylov convergence behaviour. Note that we have used standard Dirichlet boundary conditions, as opposed to the absorbing boundary conditions used in ; however, the conclusions are identical. The large difference in the number of Krylov iterations required is due to the indefinite nature of the problem, reaching a maximum around σ = − 4 ∕ h 2 (where the problem is heavily indefinite and thus hard to solve) and causing the number of iterations to suddenly drop for values of σ around − 8 ∕ h 2 , that is, where the 2D problem turns negative definite (and thus again easy to solve), as Figure (b) clearly illustrates (see colours).…”

“…Krylov convergence. Additionally, we briefly discuss the similarity between our results and the observations made in [22]. We conclude the paper with a short review of the conclusions in Section 4.…”

A new level-dependent coarse grid correction scheme for indefinite Helmholtz problems

“…Note that for all methods, good scalability in function of the wavenumber k (referred to as ‘ k ‐scalability’) is de facto not guaranteed. The rising computational cost of MG–GMRES with ECS boundaries in function of k was explained in by pointing out that the convergence rate behaves asymptotically when nearing k 2 = 4 ∕ h 2 (independently of the problem dimension). A comparable argument explains the similarly poor k ‐scalability for the level‐dependent MG scheme.…”

Multigrid methods

“…In , Reps et al analyze the Krylov convergence rate of a Helmholtz problem preconditioned with multigrid, which is applied to the Helmholtz problem formulated on a complex contour and uses polynomial smoothers at each level. For a one‐dimensional model, it is shown that the Krylov convergence rate of the continuous problem is independent of the wave number.…”