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Preconditioning of GMRES for Helmholtz problems with quasimodes

Victorita Dolean, Pierre Marchand, Axel Modave, Timothée Raynaud

TL;DR

The paper addresses slow GMRES convergence for high-frequency Helmholtz problems caused by quasimodes within domain decomposition. It develops a GMRES convergence bound based on harmonic Ritz values to explain non-linear residual decay and stagnation when small eigenvalues appear due to quasimodes. The authors propose a robust preconditioning strategy that combines ORAS domain decomposition with deflation using approximate eigenvectors and augments it with coarse-space DtN/GenEO methods, showing that deflation removes plateaus while coarse spaces accelerate convergence. Numerical experiments in a 2D scattering setup demonstrate plateaus aligned with HR values near quasimode eigenvalues and their removal through deflation, validating the approach for large-scale, high-frequency Helmholtz problems. This work provides a practical framework for achieving reliable GMRES performance in resonant Helmholtz applications.

Abstract

Finite element methods are effective for Helmholtz problems involving complex geometries and heterogeneous media. However, the resulting linear systems are often large, indefinite, and challenging for iterative solvers, particularly at high wave numbers or near resonant conditions. We derive a GMRES convergence bound that incorporates the nonlinear behavior of the relative residual and relates convergence to harmonic Ritz values. This perspective reveals how small eigenvalues associated with quasimodes can hinder convergence, and when they cease to have an effect. These phenomena occur in domain decomposition, and we illustrate them through numerical experiments. We also combine domain decomposition methods with deflation techniques using (approximate) eigenvectors tailored to resonant regimes. Their impact on GMRES performance is evaluated.

Preconditioning of GMRES for Helmholtz problems with quasimodes

TL;DR

The paper addresses slow GMRES convergence for high-frequency Helmholtz problems caused by quasimodes within domain decomposition. It develops a GMRES convergence bound based on harmonic Ritz values to explain non-linear residual decay and stagnation when small eigenvalues appear due to quasimodes. The authors propose a robust preconditioning strategy that combines ORAS domain decomposition with deflation using approximate eigenvectors and augments it with coarse-space DtN/GenEO methods, showing that deflation removes plateaus while coarse spaces accelerate convergence. Numerical experiments in a 2D scattering setup demonstrate plateaus aligned with HR values near quasimode eigenvalues and their removal through deflation, validating the approach for large-scale, high-frequency Helmholtz problems. This work provides a practical framework for achieving reliable GMRES performance in resonant Helmholtz applications.

Abstract

Finite element methods are effective for Helmholtz problems involving complex geometries and heterogeneous media. However, the resulting linear systems are often large, indefinite, and challenging for iterative solvers, particularly at high wave numbers or near resonant conditions. We derive a GMRES convergence bound that incorporates the nonlinear behavior of the relative residual and relates convergence to harmonic Ritz values. This perspective reveals how small eigenvalues associated with quasimodes can hinder convergence, and when they cease to have an effect. These phenomena occur in domain decomposition, and we illustrate them through numerical experiments. We also combine domain decomposition methods with deflation techniques using (approximate) eigenvectors tailored to resonant regimes. Their impact on GMRES performance is evaluated.

Paper Structure

This paper contains 5 sections, 1 theorem, 15 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. GMRES convergence
  3. Domain decomposition preconditioning with deflation
  4. Numerical investigations
  5. Conclusion

Key Result

theorem 1

Let $\mathbf{A}$ be a non-singular and diagonalizable matrix. Let $J \in [1, l]$. Consider: Then, for any $m > 0$ such that $l + m < N$, the local relative residual satisfies with $\kappa(\lambda_i) := \|\hat{\mathbf{v}}_i\|_2\|\mathbf{v}_i\|_2$, where $\hat{\mathbf{v}}_i$ and $\mathbf{v}_i \in \mathbb{C}^{n}$ are the left and right eigenvectors associated to $\lambda_i\in\sigma(\mathbf{A})$, wi

Figures (3)

  • Figure 1: Left: real part of the solution of \ref{['eq:helmholtzPB']}. Right: domain decomposition $(\mathcal{S}=16)$ with the mesh.
  • Figure 2: GMRES convergence with ORAS preconditioning for different number of subdomains $\mathcal{S}$ (left). Spectrum of $\mathbf{A}\mathbf{M}_{\mathrm{oras}}^{-1}$ and HR values trajectories for $\mathcal{S}=16$ (right).
  • Figure 3: GMRES convergence with DtN CS, with and without additional deflation contribution for $k=102.11$ and $\mathcal{S}=16$.

Theorems & Definitions (4)

  • definition 1: Quasimodes MarchandGalkowskiEtAl2022AGH
  • definition 2: Harmonic Ritz (HR) values
  • theorem 1: See DoleanMarchandEtAl2025CAG
  • definition 3