  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.
