
CivilComp Proceedings ISSN 17593433
CCP: 90 PROCEEDINGS OF THE FIRST INTERNATIONAL CONFERENCE ON PARALLEL, DISTRIBUTED AND GRID COMPUTING FOR ENGINEERING
Edited by:
Paper 6 One and TwoLevel Domain Decomposition Methods for Nonlinear Problems
L. Badea Institute of Mathematics, Romanian Academy, Bucharest, Romania
Full Bibliographic Reference for this paper
L. Badea, "One and TwoLevel Domain Decomposition Methods for Nonlinear Problems", in , (Editors), "Proceedings of the First International Conference on Parallel, Distributed and Grid Computing for Engineering", CivilComp Press, Stirlingshire, UK, Paper 6, 2009. doi:10.4203/ccp.90.6
Keywords: domain decomposition methods, nonlinear variational inequalities, fixedpoint problems, quasivariational inequalities, multigrid and multilevel methods, contact problems with friction, nonlinear obstacle problems.
Summary
In this paper we synthesize the results in [ 1, 2, 3, 4, 5, 6] concerning the convergence rate of the one and
twolevel 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,
 quasivariational 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 twolevel 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 twolevel
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 twolevel methods
with numerical experiments for the solution of the twoobstacle
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 twolevel method than in the case of
the onelevel method.
References
 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, 3142, 2003.
 2
 L. Badea, "Convergence rate of a Schwarz multilevel method for the constrained minimization of nonquadratic functionals", SIAM J. Numer. Anal., 44(2), 449477, 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, 427434, 2008. doi:10.1007/9783540751991_54
 4
 L. Badea, "Schwarz methods for inequalities with contraction operators", Journal of Computational and Applied Mathematics, 215(1), 196219, 2008. doi:10.1016/j.cam.2007.04.004
 5
 L. Badea, R. Krause, "One and twolevel multiplicative Schwarz methods for variational and quasivariational 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 twolevel multiplicative Schwarz methods for variational and quasivariational 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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