Analysis of preconditioned Picard iterations for the Navier-Stokes equations

Abstract

Mixed finite element formulations of fluid flow problems lead to large systems of equations of saddle-point type for which iterative solution methods are mandatory for reasons of efficiency. A successful approach in the design of solution methods takes into account the structure of the problem; in particular, it is well-known that an efficient solution can be obtained if the associated Schur complement problem can be solved efficiently and robustly. In this work we analyze a preconditioner for the Schur complement for the Oseen problem which was introduced in Numerical Analysis Report No. 99/06 We show that the spectrum of the preconditioned system is independent of the mesh parameter; moreover, we demonstrate that the number of GMRES iterations grows like the square-root of the Reynolds number for steady-state problems, while for time-dependent problems this dependence becomes negligible. In both the steady-state and time-dependent case the performance is mesh-independent.