Block preconditioning for saddle point systems with indefinite (1, 1) block

Abstract

We investigate the solution of linear systems of saddle point type with an indefinite (1, 1) block by preconditioned iterative methods. Our main focus is on block matrices arising from eigenvalue problems in incompressible fluid dynamics. A block triangular preconditioner based on an augmented Lagrangian formulation is shown to result in fast convergence of the GMRES iteration for a wide range of problem and algorithm parameters. Some theoretical estimates for the eigenvalues of the preconditioned matrices are given. Inexact variants of the preconditioner are also considered.

Keywords:

• Augmented Lagrangian
• Preconditioning
• Krylov subspace methods
• Incompressible flow

Acknowledgements

The first author was supported in part by National Science Foundation grant DMS-0511336. The authors would like to thank Professor Ke Chen for his invitation to contribute the present paper.