DELFT UNIVERSITY OF TECHNOLOGY REPORT 08-09 SIMPLE-type preconditioners for the Oseen problem

In this report, we discuss block preconditioners used to solve the incompressible Navier-Stokes equations. We emphasize on the approximation of the Schur complement used in SIMPLE-type preconditioners. In the usual formulation, the Schur complement uses scaling with the diagonal of the convection diffusion matrix. A variant of SIMPLE, SIMPLER is studied. Convergence of the SIMPLER preconditioner depends on the grid size, but not on the Reynolds number. We introduce a new variant of SIMPLER (Modified SIMPLER or MSIMPLER), based on the diagonal of the velocity mass matrix as scaling matrix instead of the diagonal of the convection-diffusion matrix. With the new approximation, we observe a drastic improvement in convergence for fairly large problems. MSIMPLER shows better convergence than the well-known least-squares commutator (LSC) preconditioner which is also based on the diagonal of the velocity mass matrix.