Convergence analysis of GMRES applied to Helmholtz problems near resonances

TL;DR

This work addresses the challenge of GMRES convergence for high-frequency Helmholtz problems near resonances and quasi-resonances by developing harmonic Ritz (HR) based convergence bounds that link residual decay to how well HR values approximate near-resonant eigenvalues. It proposes deflation with physically meaningful eigenvectors and a Complex Shifted Laplacian (CSL) preconditioner, and analyzes their combined effect on convergence for both closed-cavity and open-domain scattering scenarios. Numerical experiments show pronounced stagnation plateaus caused by small eigenvalues associated with resonances, which can be removed by deflation; the CSL preconditioner further reduces iterations, and their combination yields robust performance even in restarted GMRES settings. The results offer a practical strategy for robust, scalable solvers for Helmholtz problems at high frequencies, with potential extensions to 3D and domain-decomposition-based preconditioners.

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.

Figures (8)

• Figure 1: GMRES convergence history for a cavity problem close to a resonance.
• Figure 1: Cavity benchmark. Relative $L^2$-error and number of GMRES iterations to reach the tolerance $10^{-6}$ on the relative residual as a function of $k$. The vertical lines correspond to resonance wavenumbers.
• Figure 1: Scattering benchmark. Computational domain with PML.
• Figure 2: Cavity benchmark. Convergence analysis for $k=\hat{k}=3.01\sqrt{2}\pi$ near the resonance wavenumber $k_{3,3}$. The history of the relative residual over the iterations is shown at the top, and the trajectory of the HR values is shown at the bottom. The horizontal green lines correspond to the eigenvalues of $\mathbf{A}$.
• Figure 2: Scattering benchmark. Number of GMRES iteration to reach the relative residual $10^{-6}$, and residual history for $k=\hat{k}$ in different configurations: with/without deflation, with/without preconditioning, with/without restart.

Theorems & Definitions (14)

• Definition 2.1: Harmonic Ritz (HR) values
• Proposition 2.2: Lower bound Cao1997NCB
• Remark 2.3: Ritz values
• Lemma 2.5
• Proof 1
• Theorem 2.6: First convergence bound
• Proof 2
• Remark 2.7: Link with Cao Cao1997NCB
• Lemma 2.8: Spectral projectors
• Proof 3