Smoothers, mesh dependent norms, interpolation and multigrid Original Research Article

New estimates for the nodal interpolation operator for the P1 finite element are established with respect to mesh dependent norms that are defined in terms of the smoothers in multigrid algorithms. These estimates are useful in the additive approach to the convergence of V-cycle multigrid algorithms and lead to results on the asymptotic behavior of the contraction numbers with respect to the number of smoothing steps.