Titre : Multigrid methods for constrained minimization problems

Lori Badea

Institute of Mathematics of the Romanian Academy, Bucarest, Roumanie

The domain decomposition method proposed by H. A. Schwarz in 1869 became one hundred of years later, in the ’80 years of the previous century, of a great interest. The domain decomposition methods have the capability of providing numerical solvers which are parallelizable on parallel computers. Evidently, the initial method proposed by Schwarz has now several generalizations. Moreover, in the case of the finite element spaces, by the introduction of the coarse grids, we get the multilevel methods. The multigrid methods, which initially have been viewed as independent methods, without any connection with the Schwarz method, are in fact multilevel methods where the subdomains are the supports of the nodal basis functions.

We synthesize some results concerning the convergence rate of the multilevel and multigrid methods for the constrained minimization problems. We give general convergence results (error estimations, included) for some general subspace correction algorithms in a reflexive Banach space. The Schwarz methods, including the multilevel and multigrid methods, are introduced as particular cases in which the Sobolev or finite element spaces are used. In the finite element spaces, the constants in the error estimations are explicitly written. For the one- and two-level methods they are given as functions of the overlapping and mesh parameters. These error estimations are similar with the familiar case of the linear equations, i.e. the convergence is global and optimal. In the case of the multigrid methods, these constants are written in function of the number of levels.

Finally, numerical experiments are presented. We verify that the convergence rates obtained by numerical tests are really in concordance with the theoretical ones. We comparatively illustrate the convergence rates of the one- and two-level methods by numerical experiments for the solution of the two-obstacle problem of a nonlinear elastic membrane. These experiments show that the number of iterations, as well as the computing time, are much less in the case of the two-level method than in the case of the one-level method.