On the convergence of a multigrid method for linear reaction-diffusion problems

In this note we consider discrete linear reaction-diffusion problems. For t he discretization a standard conforming finite element method is used. For the approximate solution of the resulting discrete problem a multigrid meth od with a damped Jacobi or symmetric Gauss-Seidel smoother is applied. We a nalyze the convergence of the multigrid V- and W-cycle in the framework of the approximation- and smoothing property. The multigrid method is shown to be robust in the sense that the contraction number can be bounded by a con stant smaller than one which does not depend on the mesh size or on the dif fusion-reaction ratio.