Smoothing Factor and Actual Multigrid Convergence

We consider the Fourier analysis of multi-grid methods for symmetric positive definite and semi-positive definite linear systems arising from the discretizations of scalar PDEs. In this framework, the smoothing factor is frequently used to estimate the potential of a multigrid approach. In this paper, the smoothing factor is related to the actual two-grid convergence rate and also to the Vcycle convergence estimate based on McCormick theory in [SIAM J. Numer.Anal., 22(1985), pp.634643]. A two-sided bound is obtained that defines an interval containing both the two-grid and V-cycle convergence rate. This interval is narrow when an additional parameter is small enough, which is a simple function of quantities available in standard Fourier analysis. From a qualitative viewpoint, it turns out that, besides the smoothing factor, the convergence mainly depends on the angle between the eigenvectors of the matrix associated with small eigenvalues and the range of the prolongation. Nice V-cycle convergence is guaranteed if the tangent of this angle is proportional to the eigenvalue, whereas nice two-grid convergence requires the tangent to be proportional only to the square root of the eigenvalue. The presented results apply to rigorous Fourier analysis for regular discrete PDEs, and also to local Fourier analysis via the discussion of semi-definite systems as may arise from the discretization of PDEs with periodic boundary conditions.