Multigrid Preconditioning for Convection-Diffusion Problems on Stretched Grids

Alison Ramage

Department of Mathematics University of Strathclyde 26 Richmond Street Glasgow G1 1XH Scotland UK


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. The large size of the linear systems arising from discretisations of practical problems, particularly in three dimensions, means that iterative solution methods are often the only feasible option. When the ratio of convection to diffusion is large, standard discretisation methods can lead to oscillatory numerical solutions if the underlying grid is not sufficiently refined in areas like those within boundary layers where the solution is rapidly changing. One obvious answer is to use stretched grids which place more points inside such layers without over-refining in other parts of the domain. Unfortunately such grids are an obvious source of ill-conditioning in the resulting linear algebra problem. Any standard iterative solver therefore has to be used in conjunction with a suitable preconditioner. Multigrid methods have been used to solve convection-diffusion problems directly, but to date most research in this area seems to be confined to the regular grid case. The aim of this talk is to discuss the idea of using of multigrid methods as preconditioners for convection-diffusion problems on stretched grids. This approach has the advantage of allowing greater flexibility in the choice of smoothing and projection/restriction operators within the multigrid algorithm. We will investigate numerically the effect of grid stretching on the constituent parts of various multigrid preconditioning strategies.