Comparison of bounds for V-cycle multigrid

We consider multigrid methods with V-cycle for symmetric positive definite linear systems. We compare bounds on the convergence factor that are characterized by a constant which is the maximum over all levels of an expression involving only two consecutive levels. More particularly, we consider the classical bound by Hackbusch, a bound by McCormick, and a bound obtained by applying the successive subspace correction convergence theory with so-called a-orthogonal decomposition. We show that the constants in these bounds are closely related, and hence that these analyses are equivalent from the qualitative point of view. From the quantitative point of view, we show that the bound due to McCormick is always the best one. We also show on an example that it can give satisfactory sharp prediction of actual multigrid convergence.