Aggregation-based algebraic multigrid for convection-diffusion equations

摘要

We consider the iterative solution of large sparse linear systems arising from the upwind finite difference discretization of convection-diffusion equations. The system matrix is then an M-matrix with nonnegative row sum, and, further, when the convective flow has zero divergence, the column sum is also nonnegative, possibly up to a small correction term. We investigate aggregation-based algebraic multigrid methods for this class of matrices. A theoretical analysis is developed for a simplified two-grid scheme with one damped Jacobi postsmoothing step. An uncommon feature of this analysis is that it applies directly to problems with variable coefficients; e.g. to problems with recirculating convective flow. On the basis of this theory, we develop an approach in which a guarantee is given on the convergence rate thanks to an aggregation algorithm that allows an explicit control of the location of the eigenvalues of the preconditioned matrix. Some issues that remain beyond the analysis are discussed in the light of numerical experiments, and the efficiency of the method is illustrated on a sample of large two- and three-dimensional problems with highly varying convective flow. © 2012 Society for Industrial and Applied Mathematics.