Field of values analysis of preconditioners for the Helmholtz equation in lossy media

TL;DR

This paper analyzes the convergence of GMRES with various preconditioners for the Helmholtz equation in lossy media, using field of values analysis to account for non-normality and inexact preconditioning.

Contribution

It introduces a field of values analysis for preconditioned Helmholtz systems in lossy media, including inexact and two-level preconditioners, considering media properties not previously analyzed.

Findings

Convergence bounds for GMRES with Laplace and inexact Laplace preconditioners.

Extension of analysis to media with losses not previously considered.

Insights into the impact of media properties on preconditioner effectiveness.

Abstract

In this paper, we analyze the convergence of the preconditioned GMRES method for the first order finite element discretizations of the Helmholtz equation in media with losses. We consider a Laplace preconditioner, an inexact Laplace preconditioner and a two-level preconditioner. Our analysis is based on bounding the field of values of the preconditioned system matrix in the complex plane. The analysis takes the non-normal nature of the linear system naturally into account and allows us to easily consider certain type of inexact Laplace preconditioners via a perturbation argument. For the two-level preconditioner, our convergence analysis takes into account a media, which has not been considered in previous works.