Maximum norm convergence of multigrid methods for elliptic boundary value problems

We consider multigrid methods applied to standard linear finite element discretizations of linear elliptic boundary value problems in 2D. In the multigrid method damped Jacobi or damped Gauss-Seidel is used as a smoother. We prove that the two-grid method with v pre-smoothing iterations has a contraction number with respect to the maximum norm that is (asymptotically) bounded by Cv-~ I1n hkl 2 , with hk a suitable mesh size parameter. Moreover, it is shown that this bound is sharp in the sense that a factor lIn hkl is necessary.