Some observations on multigrid convergence for convection-diffusion equations

This paper is concerned with the convergence behaviour of multigrid methods for two-dimensional discrete convection-diffusion equations. In Elman and Ramage (BIT 46: 283-299, 2006), we showed that for constant coefficient problems with grid-aligned flow and semiperiodic boundary conditions, the two-grid iteration matrix can be reduced via a set of orthogonal transformations to a matrix containing individual 4 x 4 blocks, enabling a trivial computation of the norm of the iteration matrix. Here we use a similar Fourier analysis technique to investigate the individual contributions from the smoothing and approximation property matrices which form the basis of many standard multigrid analyses. As well as the theoretical results in the semiperiodic case, we present numerical results for a corresponding Dirichlet problem and examine the correlation between the two cases.