An Efficient Iterative Method for the Generalized Stokes Problem

TL;DR: A novel method to obtain a preconditioned linear system from the original one which is then solved by an iterative method, which generates a basis for the velocity space and solves a reduced system which is symmetric and positive definite.

Abstract: The generalized Stokes problem, which arises frequently in the simulation of time-dependent 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.