WIAS Preprint No. 2408, (2017)

An assessment of solvers for saddle point problems emerging from the incompressible Navier--Stokes equations


  • Ahmed, Naveed
    ORCID: 0000-0002-9322-0373
  • Bartsch, Clemens
  • John, Volker
    ORCID: 0000-0002-2711-4409
  • Wilbrandt, Ulrich

2010 Mathematics Subject Classification

  • 65N22


  • Linear saddle point problems, inf-sup stable pairs of finite element spaces, UMFPACK, flexible GMRES, coupled multigrid preconditioners with Vanka smoother, Least Squares Commutator preconditioners




Efficient incompressible flow simulations, using inf-sup stable pairs of finite element spaces, require the application of efficient solvers for the arising linear saddle point problems. This paper presents an assessment of different solvers: the sparse direct solver UMFPACK, the flexible GMRES (FGMRES) method with different coupled multigrid preconditioners, and FGMRES with Least Squares Commutator (LSC) preconditioners. The assessment is performed for steady-state and time-dependent flows around cylinders in 2d and 3d. Several pairs of inf-sup stable finite element spaces with second order velocity and first order pressure are used. It turns out that for the steady-state problems often FGMRES with an appropriate multigrid preconditioner was the most efficient method on finer grids. For the time-dependent problems, FGMRES with LSC preconditioners that use an inexact iterative solution of the velocity subproblem worked best for smaller time steps.

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