On GSOR, the Generalized Successive Overrelaxation Method for Double Saddle-Point Problems

TL;DR

This paper introduces the GSOR method for solving complex saddle-point problems, providing convergence analysis, spectral bounds for preconditioners, and demonstrating effectiveness through numerical experiments.

Contribution

The paper develops the GSOR method for block saddle-point problems, offering new convergence conditions and spectral bounds, with practical validation on fluid dynamics models.

Findings

GSOR converges under derived conditions.

Spectral bounds for preconditioned matrices are sharp.

Numerical tests confirm GSOR's effectiveness.

Abstract

We consider the generalized successive overrelaxation (GSOR) method for solving a class of block three-by-three saddle-point problems. Based on the necessary and sufficient conditions for all roots of a real cubic polynomial to have modulus less than one, we derive convergence results under reasonable assumptions. We also analyze a class of block lower triangular preconditioners induced from GSOR and derive explicit and sharp spectral bounds for the preconditioned matrices. We report numerical experiments on test problems from the liquid crystal director model and the coupled Stokes-Darcy flow, demonstrating the usefulness of GSOR.