Multigrid Convergence for Discretizations of Singular Perturbation Problems with Grid-Aligned Flow

Abstract

We use Fourier analysis to examine multigrid convergence for a variety of discretizations of the two-dimensional convection-diffusion equation with grid-aligned flow velocity. Emphasis is placed on the problem with a small diffusion coefficient. We consider three distinct discretizations of the problem: centered differencing, standard upwind differencing, and a generalization to two dimensions of the El-Mistikawy and Werle discretization (EMW). The analysis covers the effectiveness of block-Jacobi relaxation for the problem and stability (as the diffusion coefficient $\epsilon \to 0$) of the discretizations. In stark contrast to the problem with general (non-grid-aligned) flow, we obtain V -cycle rates independent of meshwidth h for central differencing, as well as for the other discretizations. A striking similarity is found between tandard upwinding and the second-order accurate generalization of EMW. The latter is also found to be superior to central differencing in terms of stability. Superior multigr...