Schur complement based preconditioners for twofold and block tridiagonal saddle point problems
Mingchao Cai, Guoliang Ju, Jingzhi Li
TL;DR
This work examines Schur-complement–based preconditioners for twofold and block-tridiagonal saddle point systems. It develops two broad families: nested Schur complement preconditioners and additive-type preconditioners after matrix permutation, and analyzes their spectral properties to establish positively stable preconditioned operators under appropriate sign choices. The theory extends to $n$-tuple block systems, deriving polynomial bounds on the preconditioned spectra and comparing the effectiveness of nested versus additive approaches. Numerical experiments on a 3-field Biot model validate that positively stable preconditioners offer superior convergence, especially when Schur complements are treated inexactly, with additive versions often outperforming nested ones in GMRES settings.
Abstract
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.