The development of efficient numerical solution techniques for convection-diffusion problems is an important area of current research in the field of iterative methods. As well as being of interest in their own right, convection-diffusion problems are closely linked to the Navier-Stokes equations governing incompressible fluid flow which are widely applicable in industrial settings. One possible approach which has been successfully applied in practice is to use a multigrid method, either alone or as preconditioner to an iterative solver. However, the development of related convergence analysis for the convection-diffusion problem has to date been limited. In addition, much of the published theory in the area is very technical and can be hard for the non-expert to interpret. In this talk we will present a matrix-based Fourier analysis of multigrid convergence factors for a two-dimensional model convection-diffusion equation. We will demonstrate the technique using a semiperiodic model problem and show that these results are strongly correlated with the properties of the iteration matrix arising from (more practically relevant) Dirichlet problems.
This work is in collaboration with Howard Elman, Maryland