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

TL;DR

The paper advances GSOR for a three-by-three block double saddle-point system with SPD blocks $A$ and $D$, full-row-rank $B$, and matrix $C$. It extends GSOR with three parameters $(\omega,\tau,\theta)$ and derives convergence via real cubic polynomial root criteria, along with a block lower triangular preconditioner whose spectrum is sharply bounded. Theoretical results include explicit parameter ranges ensuring convergence and detailed eigenvalue bounds for the preconditioned operator, plus practical demonstrations on liquid crystal director and Stokes–Darcy problems showing superior performance of GSOR with the proposed preconditioner over Uzawa-like methods and GBSOR in large-scale settings. The work provides a scalable framework for solving challenging double saddle-point problems in continuum mechanics and porous media, with clear guidance on parameter selection and a path toward automatic tuning via future research.

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.

Figures (4)

• Figure 1: \newlabelfig:plots0 Top left: The region of parameter values for which GSOR satisfies ${\rm Res}\le 10^{-8}$ within $5,000$ iterations. Other plots: Characteristic curves for the number of iterations versus parameters $\omega$, $\tau$ and $\theta$ for GSOR with $\omega=1$ (top right), $\tau = 1$ (bottom left), and $\theta = 1$ (bottom right). All plots are for saddle-point systems from the liquid crystal directors model with $n=3069$, $m=p=1023$.
• Figure 2: Eigenvalue distributions of the original matrix and the GSOR preconditioned matrices for saddle-point systems from the liquid crystal directors model with $n=3069,m=p=1023$.
• Figure 3: \newlabelfig:st-plots0 Top left: The region of parameter values for which GSOR satisfies ${\rm Res}\le 10^{-8}$ within $5,000$ iterations. Other plots: Characteristic curves for the number of iterations versus parameters $\omega$, $\tau$ and $\theta$ for GSOR with $\omega=1$ (top right), $\tau = 1$ (bottom left), and $\theta = 1$ (bottom right). All plots are for saddle-point systems from the mixed Stokes-Darcy model with $n=578$, $m=81$, $p=289$.
• Figure 4: Eigenvalue distributions of the original matrix and the GSOR preconditioned matrices for saddle-point systems from the mixed Stokes-Darcy model with $n=578,m=81,p=289$.

Theorems & Definitions (11)

• Lemma 3.1
• Lemma 3.2
• Theorem 3.1
• Proof 1
• Remark 3.2
• Remark 3.3
• Remark 3.4
• Theorem 4.1
• Proof 2
• Remark 4.2