We consider several preconditioners for the pressure Schur complement of the discrete
steady Oseen problem. Two of the preconditioners are well known from the literature and the other
is new. Supplemented with an appropriate approximate solve for an auxiliary velocity subproblem
these approaches give rise to a family of the block preconditioners for the matrix of the discrete
Oseen system. In the paper we critically review possible advantages and difficulties of using various
Schur complement preconditioners. We recall existing eigenvalue bounds for preconditioned Schur
complement and prove such with the newly proposed preconditioner. These bounds hold both for
LBB stable and stabilized finite elements. Results of numerical experiments for several model 2D
and 3D problems are presented. In experiments we use LBB stable finite element methods on
uniform triangular and tetrahedral meshes. One particular conclusion is that in spite of essential
improvement in comparison with /simple`` scaled mass-matrix preconditioners in certain cases, none
of the considered approaches provides satisfactory convergence rates in the case of small viscosity
coe±cients and sufficiently complex (e.g., circulating) advection vector field.
Key words. Oseen equations, saddle-point problems, finite elements, iterative methods, preconditioning, pressure Schur complement, Navier{Stokes equations
AMS subject classifications.