GMRES Convergence Analysis for Nonsymmetric Saddle-Point Systems When the Field of Values Contains the Origin

TL;DR

This paper analyzes the convergence of GMRES for nonsymmetric saddle-point systems using field-of-values analysis, providing new conditions for convergence that apply to certain preconditioners and highlighting limitations related to skew-symmetric parts.

Contribution

It introduces a novel FOV analysis for preconditioned saddle-point systems, extending applicability to new block preconditioners and establishing convergence conditions independent of matrix size.

Findings

Convergence is guaranteed when the field of values contains the origin.

Applicable to certain block preconditioners with small skew-symmetric parts.

Numerical results support the theoretical analysis.

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.