The problem of finding good preconditioners for the numerical solution of a certain important class of indefinite linear systems is considered. These systems are of a saddle point structure
In Constraint preconditioning for indefinite linear systems,
SIAM J. Matrix Anal. Appl. 21 (2000),
Keller, Gould and Wathen analyzed the idea of
using constraint preconditioners that have a
specific 2 by 2 block structure for the case of
being zero. We
shall extend this idea by allowing the (2,2) block to be
non-zero. Results concerning the spectrum and form of the
eigenvectors are presented, as are numerical results to validate
our conclusions.
We will also introduce the idea of
implicit-factorization constraint preconditioners which allow us
to efficiently carry out the required preconditioning steps.