Preconditioners for Saddle Point Problems Arising in Computational Fluid Dynamics

Discretization and linearization of the incompressible Navier-Stokes equations leads to linear algebraic systems in which the coefficient matrix has the form of a saddle point problem ( F B^T ) (u) = (f) (1) ( B 0 ) (p) (g) In this paper, we describe the development of efficient and general iterative solution algorithms for this class of problems. We review the case where (1) arises from the steady-state Stokes equations and show that solution methods such as the Uzawa algorithm lead naturally to a focus on the Schur complement operator BF^{-1}B^T together with efficient strategies of applying the action of F^{-1} to a vector. We then discuss the advantages of explicitly working with the coupled form of the block system (1). Using this point of view, we describe some new algorithms derived by developing efficient methods for the Schur complement systems arising from the Navier-Stokes equations, and we demonstrate their effectiveness for solving both steady-state and evolutionary problems. (Also referenced as UMIACS-TR-2001-88