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One- and two-level Schwarz methods for variational inequalities of the second kind and their application to frictional contact

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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References

  1. 1

    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)

    Google Scholar 

  2. 2

    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)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3

    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

    MathSciNet  MATH  Article  Google Scholar 

  4. 4

    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

    Google Scholar 

  5. 5

    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)

  6. 6

    Ciarlet P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  7. 7

    Cocu M.: Existence of solutions of Signorini problems with friction. Int. J. Eng. Sci. 5, 567–575 (1984)

    MathSciNet  Article  Google Scholar 

  8. 8

    Duvaut G., Lions J.L.: Inequalities in Mechanics and Physics. Springer, Berlin (1976)

    MATH  Book  Google Scholar 

  9. 9

    Ekeland I., Temam R.: Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976)

    MATH  Google Scholar 

  10. 10

    Glowinski R., Lions J.L., Trémolières R.: Analyse numérique des inéquations variationnelles. Dunod, Paris (1976)

    MATH  Google Scholar 

  11. 11

    Hlavá ček I., Haslinger J., JNečas J., Lovišek J.: Solution of Variational Inequalities in Mechanics. Springer, Berlin (1988)

    Book  Google Scholar 

  12. 12

    Kikuchi N., Oden J.: Contact Problems in Elasticity. SIAM, Philadelphia (1988)

    MATH  Book  Google Scholar 

  13. 13

    Kornhuber R., Krause R.: Adaptive multigrid methods for Signorini’s problem in linear elasticity. Comput. Vis. Sci. 4, 9–20 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14

    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)

    MATH  Article  Google Scholar 

  15. 15

    Lions J.L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris (1969)

    MATH  Google Scholar 

  16. 16

    Lions J.L., Magenes E.: Problèmes aux limites non homogènes et applications. Dunod, Paris (1968)

    MATH  Google Scholar 

  17. 17

    Mandel J.: A multilevel iterative method for symmetric, positve definite linear complementary problems. Appl. Math. Opt. 11, 77–95 (1984)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18

    Mandel J.: Etude algébrique d’une méthode multigrille p. quelques problémes de frontiére libre. C. R. Acad. Sci. Ser. I 298, 469–472 (1984)

    MathSciNet  MATH  Google Scholar 

  19. 19

    Radoslovescu Capatina A., Cocu M.: Internal approximation of quasi-variational inequalities. Numer. Math. 59, 385–398 (1991)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20

    Toselli A., Widlund O.: Domains Decomposition Methods—Algorithms and Theory. Springer, Berlin (2005)

    Google Scholar 

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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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Mathematics Subject Classification (2000)

  • 65N55
  • 65N30
  • 65J15