The generalized Stokes problem, which arises frequently in the simulat
ion of time-dependent Navier-Stokes equations for incompressible fluid
flow, gives rise to symmetric linear systems of equations. These syst
ems are indefinite due to a set of linear constraints on the velocity,
causing difficulty for most preconditioners and iterative methods. Th
is paper presents a novel method to obtain a preconditioned linear sys
tem from the original one which is then solved by an iterative method.
This new method generates a basis for the velocity space and solves a
reduced system which is symmetric and positive definite. Numerical ex
periments indicating superior convergence compared to existing methods
are presented. A natural extension of this method to elliptic problem
s is also proposed, along with theoretical bounds on the rate of conve
rgence, and results of experiments demonstrating robust and effective
preconditioning.