On maximum norm convergence of multigrid methods for elliptic boundary value problems

Multigrid methods applied to standard linear finite element discretizations of linear elliptic boundary value problems in two dimensions are considered. In the multigrid method, damped Jacobi or damped Gauss-Seidel is used as a smoother. It is proven that the two-grid method with v pre-smoothing interations has a contraction number with respect to the maximum norm that is (asymptotically) bounded by Cv-1/2|lnhk|2, with hk a suitable mesh size parameter. Moreover, it is shown that this bound is sharp in the sense that a factor |ln hk| is necessary.