A Simplified Analysis of the Multigrid V-Cycle as a Fast Elliptic Solver

For special model problems, Fourier analysis gives exact convergence rates for the two-grid multigrid cycle and, for more general problems, provides estimates of the two-grid convergence rates via local mode analysis. A method is presented for obtaining multigrid convergence rate estimates for cycles involving more than two grids -- using essentially the same analysis as for the two-grid cycle. For the simple case of the V-cycle used as a fast Laplace solver on the unit square, the k-grid convergence rate bounds obtained by this method are sharper than the bounds predicted by te variational theory. Both theoretical justification and experimental evidence are presented.