Convergence analysis of GMRES applied to Helmholtz problems near resonances

TL;DR

This paper investigates how the convergence of GMRES for Helmholtz problems near resonances is affected by eigenvalue approximation, deflation, and preconditioning, providing theoretical bounds and numerical insights.

Contribution

It extends convergence bounds for GMRES, analyzes the role of eigenvalue approximation and deflation, and applies these methods to near-resonant Helmholtz problems.

Findings

Eigenvalue approximation by harmonic Ritz values influences convergence.

Deflation combined with Complex Shifted Laplacian preconditioning improves convergence.

Numerical examples illustrate the convergence behavior near resonant frequencies.

Abstract

In this work we study how the convergence rate of GMRES is influenced by the properties of linear systems arising from Helmholtz problems near resonances or quasi-resonances. We extend an existing convergence bound to demonstrate that the approximation of small eigenvalues by harmonic Ritz values plays a key role in convergence behavior. Next, we analyze the impact of deflation using carefully selected vectors and combine this with a Complex Shifted Laplacian preconditioner. Finally, we apply these tools to two numerical examples near (quasi-)resonant frequencies, using them to explain how the convergence rate evolves.