On GSOR, the Generalized Successive Overrelaxation Method for Double Saddle-Point Problems
Na Huang, Yu‐Hong Dai, Dominique Orban, Michael A. Saunders · SIAM Journal on Scientific Computing · 2023
Abstract. 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.