Fourier analysis of multigrid methods for general systems of PDEs

Most iteration methods for solving boundary value problems can be viewed as approximations of a time-dependent differential equation. In this paper we show that the multigrid method has the effect of increasing the time-step for the smooth part of the solution leading back to an increase of the convergence rate. For the nonsmooth part the convergence is an effect of damping. Fourier analysis is used to find the relation between the convergence rate for multigrid methods and singlegrid methods. The analysis is performed for general partial differential equations and an arbitrary number of grids. The difference in the behavior of the iterations between first- and second-order equations is discussed. The theoretical results are confirmed in simple numerical experiments.