Abstract
We present and analyze subspace correction methods for the solution of variational inequalities of the second kind and apply these theoretical results to non smooth contact problems in linear elasticity with Tresca and non-local Coulomb friction. We introduce these methods in a reflexive Banach space, prove that they are globally convergent and give error estimates. In the context of finite element discretizations, where our methods turn out to be one- and two-level Schwarz methods, we specify their convergence rate and its dependence on the discretization parameters and conclude that our methods converge optimally. Transferring this results to frictional contact problems, we thus can overcome the mesh dependence of some fixed-point schemes which are commonly employed for contact problems with Coulomb friction.
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Badea L. et al.: Convergence rate of a multiplicative Schwarz method for strongly nonlinear inequalities. In: Barbu, V. (eds) Analysis and Optimization of Differential Systems, pp. 31–42. Kluwer, Dordrecht (2003)
Badea L.: Convergence rate of a Schwarz multilevel method for the constrained minimization of non-quadratic functionals. SIAM J. Numer. Anal. 44, 449–477 (2006)
Badea L.: Schwarz methods for inequalities with contraction operators. J. Comp. Appl. Math. 215, 196–219 (2008). doi:10.1016/j.cam.2007.04.004
Badea L.: One- and two-level domain decomposition methods for nonlinear problems. In: Topping, B.H.V., Ivnyi, P. (eds) Proceedings of the First International Conference on Parallel, Distributed and Grid Computing for Engineering., Civil-Comp Press, Scotland (2009). doi:10.4203/ccp.90.6
Badea, L., Krause, R.: One- and two-level multiplicative Schwarz methods for variational and quasi-variational inequalities of the second kind: part I and II. Institute for Numerical Simulation, University of Bonn (2008)
Ciarlet P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)
Cocu M.: Existence of solutions of Signorini problems with friction. Int. J. Eng. Sci. 5, 567–575 (1984)
Duvaut G., Lions J.L.: Inequalities in Mechanics and Physics. Springer, Berlin (1976)
Ekeland I., Temam R.: Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976)
Glowinski R., Lions J.L., Trémolières R.: Analyse numérique des inéquations variationnelles. Dunod, Paris (1976)
Hlavá ček I., Haslinger J., JNečas J., Lovišek J.: Solution of Variational Inequalities in Mechanics. Springer, Berlin (1988)
Kikuchi N., Oden J.: Contact Problems in Elasticity. SIAM, Philadelphia (1988)
Kornhuber R., Krause R.: Adaptive multigrid methods for Signorini’s problem in linear elasticity. Comput. Vis. Sci. 4, 9–20 (2001)
Krause R.: A non-smooth multiscale method for solving frictional two-bodies contact problems in 2d and 3d with multigrid efficiency. SIAM J. Sci. Comput. 31(2), 1399–1423 (2009)
Lions J.L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris (1969)
Lions J.L., Magenes E.: Problèmes aux limites non homogènes et applications. Dunod, Paris (1968)
Mandel J.: A multilevel iterative method for symmetric, positve definite linear complementary problems. Appl. Math. Opt. 11, 77–95 (1984)
Mandel J.: Etude algébrique d’une méthode multigrille p. quelques problémes de frontiére libre. C. R. Acad. Sci. Ser. I 298, 469–472 (1984)
Radoslovescu Capatina A., Cocu M.: Internal approximation of quasi-variational inequalities. Numer. Math. 59, 385–398 (1991)
Toselli A., Widlund O.: Domains Decomposition Methods—Algorithms and Theory. Springer, Berlin (2005)
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Badea, L., Krause, R. One- and two-level Schwarz methods for variational inequalities of the second kind and their application to frictional contact. Numer. Math. 120, 573–599 (2012). https://doi.org/10.1007/s00211-011-0423-y
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DOI: https://doi.org/10.1007/s00211-011-0423-y