Globally Convergent Multigrid Method for Variational Inequalities with a Nonlinear Term
In , one- and two-level Schwarz methods have been proposed for variational inequalities with contraction operators. This type of inequalities generalizes the problems modeled by quasi-linear or semilinear inequalities. It is proved there that the convergence rates of the two-level methods are almost independent of the mesh and overlapping parameters. However, the original convex set, which is defined on the fine grid, is used to find the corrections on the coarse grid, too. This leads to a suboptimal computing complexity. A remedy can be found in adopting minimization techniques from the construction of multigrid methods for the constrained minimization of functionals. In this case, to avoid visiting the fine grid, some level convex sets for the corrections on the coarse levels have been proposed in [4, 7–10] and the review article  for complementarity problems, and in  for two two-obstacle problems. In this paper, we introduce and investigate the convergence of a new multigrid algorithm for the inequalities with contraction operators, and we have adopted the construction of the level convex sets which has been introduced in . In this way, the introduced multigrid method has an optimal computing complexity of the iterations. Also, the convergence theorems for the methods introduced in  contain a convergence condition depending on the total number of the subdomains in the decompositions of the domain. The convergence condition of a direct extension of these methods to more than two-levels will introduce an upper bound for the number of mesh levels which can be used in the method. In comparison with these methods, the convergence condition of the algorithm introduced in this paper is less restrictive and depends neither on the number of the subdomains in the decompositions of the domain nor on the number of levels. Moreover, this convergence condition is very similar with the condition of existence and uniqueness of the solution of the problem.
The author acknowledges the support of this work by “Laboratoire Euroéen Associé CNRS Franco-Roumain de Matématiques et Moélisation” LEA Math-Mode.
- 3.I. Ekeland, R. Temam, Analyse Convexe et Problèmes Variationnels (Dunod, Paris, 1974)Google Scholar
- 9.J. Mandel, A multilevel iterative method for symmetric, positive definite linear complementarity problems. Appl. Math. Optim. 11, 77–95 (1984a)Google Scholar
- 10.J. Mandel, Etude algébrique d’une méthode multigrille pour quelques problèmes de frontière libre. C.R. Acad. Sci., Ser. I 298, 469–472 (1984b)Google Scholar