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

TL;DR

This paper investigates how the convergence rate of a multigrid preconditioned Krylov method for the Helmholtz equation depends on the wave number, revealing that discretization affects convergence and that finer grids improve performance.

Contribution

It provides a detailed analysis of the wave number dependency of Krylov convergence for Helmholtz problems, highlighting the impact of discretization and spectral properties.

Findings

Krylov convergence rate is wave number independent in the continuous model.

Discretization causes deviations in convergence due to spectral pitchfork.

Finer grids lead to convergence rates approaching the continuous bound.

Abstract

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 GMRES 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 due to 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.