Schur Complement Based Preconditioners for Twofold and Block Tridiagonal Saddle Point Problems (Communications in Mathematical Research, 2024)
In this paper, we consider using Schur complements to design preconditioners for twofold and block tridiagonal saddle point problems. Onetype of the preconditioners are based on the nested (or recursive) Schur complement, the other is based on an additive type Schur complement after permuting the original saddle point systems. We analyze different preconditionersincorporating the exact Schur complements. We show that some of them willlead to positively stable preconditioned systems if proper signs are selected infront of the Schur complements. These positive-stable preconditioners outperform other preconditioners if the Schur complements are further approximatedinexactly. Numerical experiments for a 3-field formulation of the Biot modelare provided to verify our predictions.
Keywords: Schur complement; block tridiagonal systems; positively stable preconditioners; Routh-Hurwitz stability criterion.