Robust convergence of multilevel algorithms for convection-diffusion equations

A new approach is developed to analyze convergence of multilevel algorithms for convection-diffusion equations. This approach uses a multilevel recursion formula, which can be applied to a variety of nonsymmetric problems. Here, the recursion formula is applied to a robust multilevel algorithm for convection-diffusion equations with convection in the x- or y-direction. The multilevel algorithm uses semicoarsening, line relaxation, and prewavelets. The convergence rate is proved to be less than 0.18 independent of the size of the convection term and the number of unknowns. The assumptions allow the convection term to have a turning point, so that an interior layer can appear in the solution of the convection-diffusion equation. The computational cost of the multilevel cycle is about O(N log N) independent of the size of the convection term, where N is the number of unknowns. It is proved that O(log N) multilevel cycles starting from the initial guess 0 lead to an O(N(-2)) algebraic error with respect to the L(infinity) norm, independent of the size of the convection term.