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 nu pre-smoothing iterations has a contraction number with respect to the maximum norm that is (asymptotically) bounded by Cnu-1/2 \ln h(k)\2, with h(k); a suitable mesh size parameter, Moreover, it is shown that this bound is sharp in the sense that a factor \ln h(k)\ is necessary.