GMRES Convergence Analysis for Nonsymmetric Saddle-Point Systems When the Field of Values Contains the Origin
Abstract
We present a field-of-values (FOV) analysis for preconditioned nonsymmetric saddle-point linear systems, where zero is included in the field of values of the matrix. We rely on recent results of Crouzeix and Greenbaum [Spectral sets: numerical range and beyond. SIAM Journal on Matrix Analysis and Applications, 40(3):1087-1101, 2019], showing that a convex region with a circular hole is a spectral set. Sufficient conditions are derived for convergence independent of the matrix dimensions. We apply our results to preconditioned nonsymmetric saddle-point systems, and show their applicability to families of block preconditioners that have not been previously covered by existing FOV analysis. A limitation of our theory is that the preconditioned matrix is required to have a small skew-symmetric part in norm. Consequently, our analysis may not be applicable, for example, to fluid flow problems characterized by a small viscosity coefficient. Some numerical results illustrate our findings.
Keywords
Cite
@article{arxiv.2505.07156,
title = {GMRES Convergence Analysis for Nonsymmetric Saddle-Point Systems When the Field of Values Contains the Origin},
author = {Hao Chen and Chen Greif},
journal= {arXiv preprint arXiv:2505.07156},
year = {2026}
}