In this talk we will present a result on the asymptotic behavior of the contraction numbers of multigrid V-cycle algorithms with respect to the number of smoothing steps. This result is obtained using an additive convergence theory and new estimates for mesh-dependent norms. When combined with earlier results of Zhang, Bramble and Pasciak, it yields a generalization of the classical convergence theorem of Braess and Hackbusch to problems without full elliptic regularity.