Preconditioning for Krylov methods often relies on operator theory when mesh independent estimates are looked for. The goal of this paper is to contribute to the long development of the analysis of superlinear convergence of Krylov iterations when the preconditioned operator is a compact perturbation of the identity. Mesh independent superlinear convergence of GMRES and CGN iterations is derived for Galerkin solutions for complex non-Hermitian and noncoercive operators. The results are applied to noncoercive boundary value problems, including shifted Laplacian preconditioners for Helmholtz problems.


  1. Krylov iteration
  2. preconditioning
  3. noncoercive operators
  4. mesh independence
  5. shifted Laplace

MSC codes

  1. 65F10
  2. 65J10
  3. 65N30

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Information & Authors


Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 1057 - 1079
ISSN (online): 1095-7170


Submitted: 22 December 2021
Accepted: 22 November 2022
Published online: 27 April 2023


  1. Krylov iteration
  2. preconditioning
  3. noncoercive operators
  4. mesh independence
  5. shifted Laplace

MSC codes

  1. 65F10
  2. 65J10
  3. 65N30


Owe Axelsson passed away in June 2022. It was a privilege that we could work with him on this and previous papers, and we will never forget his unique inspiration and presence.



Owe Axelsson
Institute of Geonics AS CR, IT4 Innovations, Ostrava, Czech Republic.
Department of Applied Analysis and ELKH-ELTE Numerical Analysis and Large Networks Research Group, Eötvös Loránd University, Budapest, Hungary; and Department of Analysis, Budapest University of Technology and Economics, Hungary.
Frédéric Magoulès
CentraleSupelec, Université Paris-Saclay, France, and University of Pécs, Pécs, Hungary.

Funding Information

Funding: This research has been supported by the National Research, Development and Innovation Office (NKFIH), grants K137699 and SNN125119.

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