Compact Fourier Analysis for Designing Multigrid Methods

The convergence of multigrid methods can be analyzed based on a Fourier analysis of the method or by proving certain inequalities that have to be fulfilled by the smoother and by the coarse grid correction separately. Here, we analyze the multigrid method for the constant coefficient Poisson equation with a compact Fourier analysis using the formalism of multilevel Toeplitz matrices and their generating functions or symbols. The Fourier analysis is applied for determining the smoothing factor and the overall error of the combined smoothing and coarse grid correction error reduction of a twogrid step (TGS) by representing the twogrid step explicitly by a symbol. If the effects of the smoothing correction and the coarse grid correction are orthogonal to each other, then, in a twogrid step, the error is removed in one step, and the twogrid method can be considered as a direct solver. If the coarse linear system is identical to the original matrix, then the same projection and smoother again make the twogrid step a direct solver. Hence, the multigrid cycle has to be applied only once with one smoothing step on each level, and therefore the whole multigrid method can be considered as a direct solver. In this paper we want to identify multigrid as a direct solver in one-dimension with a coarse system derived by linear and constant interpolation, and in two-dimensions for a modified projection related to [U. Trottenberg, C.W. Osterlee, and A. Schüller, Multigrid, Academic Press, San Diego, 2001, paragraph A.2.3]. By studying not only smoothing and coarse grid correction separately but the symbol of the full twogrid step as well, we are furthermore able to derive better convergence estimates and information on how to find efficient combinations of projection and smoother. As smoothers we consider also approximate inverse smoothers, colored smoothers that take into account the different character of grid points, and rank-reducing smoothers that coincide with the given matrix on a large number of entries. Numerical examples show the effectiveness of the new approach.