Original article

Numerische Mathematik

, Volume 90, Issue 4, pp 665-688

First online:

Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations

  • Howard C. ElmanAffiliated withDepartment of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, MD 20742, USA; e-mail: elman@cs.umd.edu
  • , David J. SilvesterAffiliated withDepartment of Mathematics, University of Manchester, Institute of Science and Technology, Manchester M601QD, UK; e-mail: djs@lanczos.ma.umist.ac.uk
  • , Andrew J. WathenAffiliated withOxford University, Computing Laboratory, Wolfson Building, Parks Road, Oxford OX13QD, UK; e-mail: Andy.Wathen@comlab.ox.ac.uk

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We examine the convergence characteristics of iterative methods based on a new preconditioning operator for solving the linear systems arising from discretization and linearization of the steady-state Navier-Stokes equations. With a combination of analytic and empirical results, we study the effects of fundamental parameters on convergence. We demonstrate that the preconditioned problem has an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers. The structure of these distributions is independent of the discretization mesh size, but the cardinality of the set of outliers increases slowly as the viscosity becomes smaller. These characteristics are directly correlated with the convergence properties of iterative solvers.

Mathematics Subject Classification (1991): 65N22