On Parameter Choice and Iterative Convergence for Stabilised Discretisations of Advection-diffusion Problems

In this work we consider the design of robust and eecient nite element approximation methods for solving advection-diiusion equations. Speciically, we consider the stabilisation of discrete approximations using uniform grids which do not resolve boundary layers, as might arise using a multi-level (or multigrid) iteration strategy to solve the discrete problem. Our analysis shows that when using SUPG (streamline-upwind) nite element methodology, there is a symbiotic relationship between`best' solution approximation and fast convergence of smoothers based on the standard GMRES iteration. We also show that stabilisation based on simple artiicial diiusion perturbation terms (an approach often advocated by multigrid practitioners) is less appealing.