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. Specifically, 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) finite 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 artificial diffusion perturbation terms (an approach often advocated by multigrid practitioners) is less appealing.