On parameter choice and iterative convergence for stabilised discretisations of advection-diffusion problems

In this work we consider the design of robust and efficient finite element approximation methods for solving advection-diffusion equations. Specifical ly, we consider the stabilisation of discrete approximations using uniform grids which do not resolve boundary layers, as might arise using a multi-le vel (or multigrid) iteration strategy to solve the discrete problem. Our an alysis shows that when using SUPG (streamline-upwind) finite element method ology, there is a symbiotic relationship between 'best' solution approximat ion and fast convergence of smoothers based on the standard GMRES iteration . We also show that stabilisation based on simple artificial diffusion pert urbation terms (an approach often advocated by multigrid practitioners) is less appealing. (C) 1999 Elsevier Science S.A. All rights reserved.