David Silvester universite de Manchester, Title: ------ A Review of Preconditioning Techniques for Steady Incompressible Flow Abstract: --------- Simulation of the motion of an incompressible fluid remains an important but very challenging problem. The resources required for accurate three-dimensional simulation of practical flows test even the most advanced computer hardware. The necessity for reliable and efficient solvers is widely recognised. Mixed finite element approximation of the underlying PDEs leads to symmetric indefinite or unsymmetric indefinite linear systems of equations. We describe a generic block preconditioning technique for such systems with the property that the eigenvalues of the preconditioned matrices are contained in intervals that are bounded independently of the mesh size. The attractive feature of our technique is that the basis of the preconditioning is a readily available building block; namely, a scalar diffusion or convection-diffusion solve based on an algebraic multigrid V-cycle. Numerical results are presented showing the effectiveness of our approach in the context of diffusion equations that arise in modelling ground-water flow in porous media that exhibit random spatial variability. We further demonstrate the effectiveness of our methodology in the context of solving steady flow problems modelled by the Navier-Stokes equations.