TY - JOUR
T1 - Schur Complement Based Preconditioners for Twofold and Block Tridiagonal Saddle Point Problems
AU - Cai , Mingchao
AU - Ju , Guoliang
AU - Li , Jingzhi
JO - Communications in Mathematical Research 
VL - 2
SP - 214
EP - 244
PY - 2024
DA - 2024/05
SN - 40
DO - http://doi.org/10.4208/cmr.2023-0051
UR - https://global-sci.org/intro/article_detail/cmr/23088.html
KW - Schur complement, block tridiagonal systems, positively stable preconditioners, Routh-Hurwitz stability criterion.
AB - <p style="text-align: justify;">In this paper, we consider using Schur complements to design preconditioners for twofold and block tridiagonal saddle point problems. One
type 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 preconditioners
incorporating the exact Schur complements. We show that some of them will
lead to positively stable preconditioned systems if proper signs are selected in
front of the Schur complements. These positive-stable preconditioners outperform other preconditioners if the Schur complements are further approximated
inexactly. Numerical experiments for a 3-field formulation of the Biot model
are provided to verify our predictions.</p>