Global convergence rate of a standard multigrid method for variational inequalities

The main purpose of this paper is to give an estimation of the global convergence rate for the standard monotone multigrid method applied to variational inequalities whose constraints are of the two-obstacle type. The numerical experiments using this method have highlighted global upper bounds of the convergence rate, but, to our knowledge, no theoretical justification exists so far. The method was introduced by Mandel in 1984 for complementarity problems and named later by Kornhuber as the standard monotone multigrid method. First, we introduce the method as a subspace correction algorithm in a reflexive Banach space, prove its global convergence and estimate the error after making some assumptions. By introducing finite element spaces, this algorithm becomes a multilevel or multigrid method. In this case, we prove that the assumptions that we made in the general theory are satisfied and write the convergence rate as a function of the number of levels. Finally, we compare our results with the estimations of the asymptotic convergence rate existing in the literature for complementarity problems.