Preconditioners for Saddle Point Problems Arising in Computational Fluid Dynamics

Discretization and linearization of the incompressible Navier-Stokesequations 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)