Keywords: domain decomposition methods, nonlinear variational inequalities, fixed-point problems, quasi-variational inequalities, multigrid and multilevel methods, contact problems with friction, nonlinear obstacle problems.

In this paper we synthesize the results in [1,2,3,4,5,6] concerning the convergence rate of the one- and two-level methods for some nonlinear problems:

• nonlinear variational inequalities,
• inequalities with contraction operators,
• variational inequalities of the second kind, with application to the contact problems with Tresca friction,
• quasi-variational inequalities, with application to the contact problems with Coulomb friction.

For all these nonlinear problems, the same proof techniques are used. In Section 2 of the paper, we give a general background in which the problems in the next sections are stated. Depending on the type of the algorithm, additive or multiplicative, we also introduce in this section some assumptions concerning the convex set. Section 3 is dedicated to the theoretical convergence results of the algorithms. We give, using the assumptions in the previous section, general convergence results (error estimations, included) for some subspace correction algorithms in a reflexive Banach space. In Section 4, the Schwarz methods, including the one and two-level methods, are introduced as particular cases in which the Sobolev or finite element spaces are used. When the subspaces are associated with a domain decomposition, we obtain the multiplicative and additive Schwarz methods. We prove that the general assumptions in Section 2 are satisfied in this case. In the finite element spaces, for the one- and two-level methods, the constants in the error estimations are explicitly written as functions of the overlapping and mesh parameters. These error estimations are similar with the familiar case of the linear equations, ie. the convergence is global and optimal. Finally, Section 5 is dedicated to numerical experiments. We verify that the convergence rates obtained by numerical tests are in accordance with the theoretical ones. We illustrate the convergence rates of the one- and two-level methods with 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.

1

L. Badea, "Convergence rate of a multiplicative Schwarz method for strongly nonlinear inequalities", in V.Barbu et al., (editors), "Analysis and optimization of differential systems", Kluwer Academic Publishers, 31-42, 2003.

2

L. Badea, "Convergence rate of a Schwarz multilevel method for the constrained minimization of non-quadratic functionals", SIAM J. Numer. Anal., 44(2), 449-477, 2006. doi:10.1137/S003614290342995X

3

L. Badea, "Additive Schwarz method for the constrained minimization of functionals in reflexive Banach spaces", in U. Langer et al., (editors), "Domain decomposition methods in science and engineering XVII", LNSE 60, Springer, 427-434, 2008. doi:10.1007/978-3-540-75199-1_54

4

L. Badea, "Schwarz methods for inequalities with contraction operators", Journal of Computational and Applied Mathematics, 215(1), 196-219, 2008. doi:10.1016/j.cam.2007.04.004

5

L. Badea, R. Krause, "One- and two-level multiplicative Schwarz methods for variational and quasi-variational inequalities of the second kind: Part I - general convergence results", INS Preprint, no. 0804, Institute for Numerical Simulation, University of Bonn, 2008.

6

L. Badea, R. Krause, "One- and two-level multiplicative Schwarz methods for variational and quasi-variational inequalities of the second kind: Part II - frictional contact problems", INS Preprint, no. 0805, Institute for Numerical Simulation, University of Bonn, 2008.

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