Fourier Analysis of Multigrid for a Model Two-Dimensional Convection-Diffusion Equation

We present a Fourier analysis of multigrid for the two-dimensional discrete convection-diffusion equation. For constant coefficient problems with grid-aligned flow and semi-periodic boundary conditions, we show that the two-grid iteration matrix can be reduced via a set of orthogonal transformations to a matrix containing individual 4×4 blocks. This enables a trivial computation of the norm of the iteration matrix demonstrating rapid convergence in the case of both small and large mesh Peclet numbers, where the streamline-diffusion discretisation is used in the latter case. We also demonstrate that these results are strongly correlated with the properties of the iteration matrix arising from Dirichlet boundary conditions. AMS subject classification (2000) 65F10, 65N22, 65N30, 65N55