MULTIGRID CONVERGENCE FOR CONVECTION-DIFFUSION PROBLEMS ON COMPOSITE GRIDS

We study the convergence behavior of the FAC (fast adaptive composite) multigrid method as applied to the solution of convection-diffusion equations discretized on composite grids in one and two dimensions. Analysis of the one-dimensional problem leads to the interpretation of two-level FAC as a direct solver. This analysis also provides important insight into the behavior of the method for the two-dimensional problem that has its flow velocity oriented in a single coordinate direction. For the latter problem we consider the use of standard upwind differencing on the coarse component of the grid, and allow the discretization type to vary on the fine component. When centered differencing is used on this fine region, we show that the behavior is very similar to that predicted by the analysis in one dimension. With a higher-order upwinding scheme used on this component, we show how to modify the discretization at the coarse-grid level in order to preserve the attractive convergence behavior predicted by the analysis.