### Abstract

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 language | English |
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Pages (from-to) | 1651-1668 |

Number of pages | 18 |

Journal | SIAM Journal on Scientific Computing |

Volume | 27 |

Issue number | 5 |

DOIs | |

State | Published - 2006 |

### Keywords

- Iterative algorithms
- Navier-Stokes
- Preconditioning

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## Cite this

*SIAM Journal on Scientific Computing*,

*27*(5), 1651-1668. https://doi.org/10.1137/040608817