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.