An Efficient Iterative Method for the Generalized Stokes Problem

The generalized Stokes problem, which arises frequently in the simulation of timedependent Navier–Stokes equations for incompressible fluid flow, gives rise to symmetric linear systems of equations. These systems are indefinite due to a set of linear constraints on the velocity, causing difficulty for most preconditioners and iterative methods. This paper presents a novel method to obtain a preconditioned linear system 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 experiments indicating superior convergence compared to existing methods are presented. A natural extension of this method to elliptic problems is also proposed, along with theoretical bounds on the rate of convergence, and results of experiments demonstrating robust and effective preconditioning.