Downwind numbering: Robust multigrid for convection-diffusion problems

In the present paper, we introduce and investigate a robust smoothing strategy for convection-diffusion problems in two and three space dimensions without any assumption on the grid structure. The main tool to obtain such a robust smoother for these problems is an ordering strategy for the grid points called ''downwind numbering'' which follows the flow direction and, combined with a Gauss-Seidel type smoother, yields robust multigrid convergence for adaptively refined grids, provided the convection field is cycle-free. The algorithms are of optimum complexity and the corresponding smoothers are shown to be robust in numerical tests.