An iterative solver for the Oseen problem and numerical solution of incompressible Navier-Stokes equations

Incompressible unsteady Navier-Stokes equations in pressure - velocity vari ables are considered. By use of the implicit and semi-implicit schemes pres ented the resulting system of linear equations can be solved by a robust an d efficient iterative method. This iterative solver is constructed for the system of linearized Navier-Stokes equations. The Schur complement techniqu e is used. We present a new approach of building a non-symmetric preconditi oner to solve a non-symmetric problem of convection-diffusion and saddle-po int type. It is shown that handling the differential equations properly res ults in constructing efficient solvers for the corresponding finite linear algebra systems. The method has good performance for various ranges of visc osity and can be used both for 2D and 3D problems. The analysis of the meth od is still partly heuristic, however, the mathematically rigorous results are proved for certain cases. The proof is based on energy estimates and ba sic properties of the underlying partial differential equations. Numerical results are provided. Additionally, a multigrid method for the auxiliary co nvection-diffusion problem is briefly discussed. Copyright (C) 1999 John Wi ley & Sons, Ltd.