Alison Ramage : March 24, 2006

Multigrid Solution of Discrete Convection-Diffusion Equations.

Abstract

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. The aim of this talk is to present a matrix-based Fourier analysis of multigrid convergence factors for the two-dimensional 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.