A new level-dependent coarsegrid correction scheme for indefinite Helmholtz problems

TL;DR

This paper introduces a novel level-dependent multigrid correction scheme for indefinite Helmholtz problems, maintaining stability and efficiency across all levels by adapting the complex shift during multigrid cycles.

Contribution

It presents a new multigrid method with a grid-dependent complex shift that ensures stability and effectiveness for solving Helmholtz equations, outperforming existing methods.

Findings

Method is stable on all levels due to adaptive complex shift.

Numerical results show competitive or superior performance.

Effective damping of oscillatory errors achieved with smoothing schemes.

Abstract

In this paper we construct and analyse a level-dependent coarsegrid correction scheme for indefinite Helmholtz problems. This adapted multigrid method is capable of solving the Helmholtz equation on the finest grid using a series of multigrid cycles with a grid-dependent complex shift, leading to a stable correction scheme on all levels. It is rigourously shown that the adaptation of the complex shift throughout the multigrid cycle maintains the functionality of the two-grid correction scheme, as no smooth modes are amplified in or added to the error. In addition, a sufficiently smoothing relaxation scheme should be applied to ensure damping of the oscillatory error components. Numerical experiments on various benchmark problems show the method to be competitive with or even outperform the current state-of-the-art multigrid-preconditioned Krylov methods, like e.g. CSL-preconditioned GMRES or BiCGStab.