GMRES-based multigrid for the complex scaled preconditoner for the indefinite Helmholtz equation

TL;DR

This paper introduces a GMRES-based multigrid preconditioner for the indefinite Helmholtz equation, improving stability and efficiency over traditional smoothers by replacing them with GMRES iterations.

Contribution

The paper proposes using GMRES as a replacement for stationary smoothers in multigrid methods, enhancing robustness for complex shifted Helmholtz problems.

Findings

Requires few GMRES iterations per level

Demonstrates improved convergence stability

Outperforms traditional polynomial smoothers

Abstract

Multigrid preconditioners and solvers for the indefinite Helmholtz equation suffer from non-stability of the stationary smoothers due to the indefinite spectrum of the operator. In this paper we explore GMRES as a replacement for the stationary smoothers of the standard multigrid method. This results in a robust and efficient solver for a complex shifted or stretched Helmholtz problem that can be used as a preconditioner. Very few GMRES iterations are required on each level to build a good multigrid method. The convergence behavior is compared to a theoretically derived stable polynomial smoother. We test this method on some benchmark problems and report on the observed convergence behavior.