Abstract | ||
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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. |
Year | DOI | Venue |
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2006 | 10.1137/040608817 | SIAM J. Scientific Computing |
Keywords | Field | DocType |
incompressible flow,preconditioning,iterative algorithm | Convergence (routing),Discretization,Mathematical optimization,Algebraic number,Commutative property,Preconditioner,Iterative method,Matrix (mathematics),Mathematical analysis,Operator (computer programming),Mathematics | Journal |
Volume | Issue | ISSN |
27 | 5 | 1064-8275 |
Citations | PageRank | References |
40 | 2.31 | 6 |
Authors | ||
5 |
Name | Order | Citations | PageRank |
---|---|---|---|
Howard C. Elman | 1 | 498 | 96.06 |
Victoria E. Howle | 2 | 103 | 10.22 |
John N. Shadid | 3 | 259 | 32.24 |
Robert Shuttleworth | 4 | 58 | 3.57 |
Ray S. Tuminaro | 5 | 447 | 38.09 |