The generalized successive overrelaxation (GSOR) method of Bai, Parlett, and Wang [Numer. Math., 102 (2005), pp. 1–38] was designed for conventional saddle-point problems. We extend GSOR to 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. 

Publication: 

SIAM Journal on Scientific Computing, Vol. 45, Iss. 5 (2023) 

http://dx.doi.org/10.1137/22M1515884 

Author: 

Na Huang 

Department of Applied Mathematics, College of Science, China Agricultural University, 100083 Beijing, China. 

Yu-Hong Dai 

LSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100190 Beijing, China. 

Email: dyh@lsec.cc.ac.cn 

Dominique Orban 

GERAD and Department of Mathematics and Industrial Engineering, Polytechnique Montréal, QC H3C 3A7, Canada. 

Michael A. Saunders 

Systems Optimization Laboratory, Department of Management Science and Engineering, Stanford University, Stanford, CA 94305 USA.