PRECONDITIONING CONVECTION DOMINATED CONVECTION-DIFFUSION PROBLEMS

This paper is concerned with the numerical solution of multi-dimensional convection dominated convection-diffusion problems. These problems are characterized by a large parameter, K, multiplying the convection terms. The goal of this work is the development and analysis of effective preconditioners for iteratively solving the large system of linear equations arising from various finite element and finite difference discretizations with grid size h. When centered finite difference schemes and standard Galerkin finite element methods are used, h must be related to K by the stability constraint, Kh less-than-or-equal-to C0, where the constant C0 is sufficiently small. A class of preconditioners is developed that significantly reduces the condition number for large K and small h. Furthermore, these preconditioners are inexpensive to implement and well suited for parallel computation. It is shown that under suitable assumptions, the number of iterations remains bounded as h --> 0 with K fixed and, at worst, grows slowly as K --> infinity. Numerical results are presented illustrating the theory. It is also shown how to apply the theoretical results to more general convection-diffusion problems and alternative discretizations (including streamline diffusion methods) that remain stable as Kh --> infinity.