A Robust Two-Level Incomplete Factorization for (Navier–)Stokes Saddle Point Matrices

Fred W. Wubs*, Jonas Thies

*Corresponding author for this work

    Research output: Contribution to journalArticleAcademicpeer-review

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    We present a new hybrid direct/iterative approach to the solution of a special class of saddle point matrices arising from the discretization of the steady incompressible Navier-Stokes equations on an Arakawa C-grid. The two-level method introduced here has the following properties: (i) it is very robust, even close to the point where the solution becomes unstable; (ii) a single parameter controls fill and convergence, making the method straightforward to use; (iii) the convergence rate is independent of the number of unknowns; (iv) it can be implemented on distributed memory machines in a natural way; (v) the matrix on the second level has the same structure and numerical properties as the original problem, so the method can be applied recursively; (vi) the iteration takes place in the divergence-free space, so the method qualifies as a "constraint preconditioner"; (vii) the approach can also be applied to Poisson problems. This work is also relevant for problems in which similar saddle point matrices occur, for instance, when simulating electrical networks, where one has to satisfy Kirchhoff's conservation law for currents.

    Original languageEnglish
    Pages (from-to)1475-1499
    Number of pages25
    JournalSIAM Journal on Matrix Analysis and Applications
    Issue number4
    Publication statusPublished - 2011


    • saddle point problem
    • indefinite matrix
    • F-matrix
    • incomplete factorization
    • grid-independent convergence
    • Arakawa C-grid
    • incompressible (Navier-)Stokes equations
    • constraint preconditioning
    • electrical networks
    • FLOW
    • ILU

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