Block preconditioners based on approximate commutators

Howard Elman, Victoria E. Howle, John Shadid, Robert Shuttleworth, Ray Tuminaro

Research output: Contribution to journalArticlepeer-review

111 Scopus citations


This paper introduces a strategy for automatically generating a block preconditioner for solving the incompressible Navier-Stokes equations. We consider the "pressure convection-diffusion preconditioners" proposed by Kay, Loghin, and Wathen [SIAM J. Sci. Comput., 24 (2002), pp. 237-256] and Silvester, Elman, Kay, and Wathen [J. Comput. Appl. Math., 128 (2001), pp. 261-279]. Numerous theoretical and numerical studies have demonstrated mesh independent convergence on several problems and the overall efficacy of this methodology. A drawback, however, is that it requires the construction of a convection-diffusion operator (denoted F p) projected onto the discrete pressure space. This means that integration of this idea into a code that models incompressible flow requires a sophisticated understanding of the discretization and other implementation issues, something often held only by the developers of the model. As an alternative, we consider automatic ways of computing F p based on purely algebraic considerations. The new methods are closely related to the "BFBt preconditioner" of Elman [SIAM J. Sci. Comput., 20 (1999), pp. 1299-1316]. We use the fact that the preconditioner is derived from considerations of commutativity between the gradient and convection-diffusion operators, together with methods for computing sparse approximate inverses, to generate the required matrix F p automatically. We demonstrate that with this strategy the favorable convergence properties of the preconditioning methodology are retained.

Original languageEnglish
Pages (from-to)1651-1668
Number of pages18
JournalSIAM Journal on Scientific Computing
Issue number5
StatePublished - 2006


  • Iterative algorithms
  • Navier-Stokes
  • Preconditioning


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