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Convergence analysis of penalty based numerical methods for constrained inequality problems

  • Weimin HanEmail author
  • Mircea Sofonea
Article
  • 32 Downloads

Abstract

This paper presents a general convergence theory of penalty based numerical methods for elliptic constrained inequality problems, including variational inequalities, hemivariational inequalities, and variational–hemivariational inequalities. The constraint is relaxed by a penalty formulation and is re-stored as the penalty parameter tends to zero. The main theoretical result of the paper is the convergence of the penalty based numerical solutions to the solution of the constrained inequality problem as the mesh-size and the penalty parameter approach zero independently. The convergence of the penalty based numerical methods is first established for a general elliptic variational–hemivariational inequality with constraints, and then for hemivariational inequalities and variational inequalities as special cases. Applications to problems in contact mechanics are described.

Mathematics Subject Classification

65N30 65N15 74M10 74M15 

Notes

References

  1. 1.
    Atkinson, K., Han, W.: Theoretical Numerical Analysis: A Functional Analysis Framework, 3rd edn. Springer, New York (2009)zbMATHGoogle Scholar
  2. 2.
    Chernov, M., Maischak, A., Stephan, E.: A priori error estimates for hp penalty bem for contact problems in elasticity. Comput. Methods Appl. Mech. Eng. 196, 3871–3880 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)zbMATHGoogle Scholar
  4. 4.
    Chouly, F., Hild, P.: On convergence of the penalty method for unilateral contact problems. Appl. Numer. Math. 65, 27–48 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Theory. Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York (2003)CrossRefzbMATHGoogle Scholar
  6. 6.
    Eck, C., Jarušek, J.: Existence results for the static contact problem with Coulomb friction. Math. Mod. Methods Appl. 8, 445–463 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976)zbMATHGoogle Scholar
  8. 8.
    Girault, V., Raviart, P.-A.: Finite Element Methods for Navier–Stokes Equations, Theory and Algorithms. Springer, Berlin (1986)CrossRefzbMATHGoogle Scholar
  9. 9.
    Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984)CrossRefzbMATHGoogle Scholar
  10. 10.
    Glowinski, R., Lions, J.-L., Trémolières, R.: Numerical Analysis of Variational Inequalities. North-Holland, Amsterdam (1981)zbMATHGoogle Scholar
  11. 11.
    Gwinner, J., Stephan, E.: Advanced Boundary Element Methods: Treatment of Boundary Value, Transmission and Contact Problems. Springer, New York (2018)CrossRefzbMATHGoogle Scholar
  12. 12.
    Han, W.: Numerical analysis of stationary variational–hemivariational inequalities with applications in contact mechanics. Math. Mech. Solids 23, 279–293 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Han, W., Migórski, S., Sofonea, M.: A class of variational–hemivariational inequalities with applications to frictional contact problems. SIAM J. Math. Anal. 46, 3891–3912 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Han, W., Migórski, S., Sofonea, M.: On a penalty based numerical method for unilateral contact problems with non-monotone boundary condition. J. Comput. Appl. Math. 356, 293–301 (2019)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Han, W., Sofonea, M.: Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, Studies in Advanced Mathematics, vol. 30. Americal Mathematical Society, RI-International Press, Providence, Somerville (2002)CrossRefzbMATHGoogle Scholar
  16. 16.
    Han, W., Sofonea, M., Barboteu, M.: Numerical analysis of elliptic hemivariational inequalities. SIAM J. Numer. Anal. 55, 640–663 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Han, W., Sofonea, M., Danan, D.: Numerical analysis of stationary variational–hemivariational inequalities. Numer. Math. 139, 563–592 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Haslinger, J., Miettinen, M., Panagiotopoulos, P.D.: Finite Element Method for Hemivariational Inequalities. Theory, Methods and Applications. Kluwer Academic Publishers, Boston (1999)CrossRefzbMATHGoogle Scholar
  19. 19.
    Hild, P., Renard, Y.: An improved a priori error analysis for finite element approximations of Signorini’s problem. SIAM J. Numer. Anal. 50, 2400–2419 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hlaváček, I., Lovíšek, J.: A finite element analysis for the Signorini problem in plane elastostatics. Apl. Mat. 22, 215–227 (1977)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Kikuchi, N., Oden, J.T.: Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia (1988)CrossRefzbMATHGoogle Scholar
  22. 22.
    Migórski, S., Ochal, A., Sofonea, M.: Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics, vol. 26. Springer, New York (2013)zbMATHGoogle Scholar
  23. 23.
    Migórski, S., Ochal, A., Sofonea, M.: A class of variational-hemivariational inequalities in reflexive Banach spaces. J. Elast. 127, 151–178 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Naniewicz, Z., Panagiotopoulos, P.D.: Mathematical Theory of Hemivariational Inequalities and Applications. Marcel Dekker Inc, New York (1995)zbMATHGoogle Scholar
  25. 25.
    Panagiotopoulos, P.D.: Hemivariational Inequalities, Applications in Mechanics and Engineering. Springer, Berlin (1993)CrossRefzbMATHGoogle Scholar
  26. 26.
    Sofonea, M., Migórski, S.: Variational-Hemivariational Inequalities with Applications. Chapman & Hall, CRC Press, Boca Raton, London (2018)zbMATHGoogle Scholar
  27. 27.
    Sofonea, M., Migórski, S., Han, W.: A penalty method for history-dependent variational–hemivariational inequalities. Comput. Math. Appl. 75, 2561–2573 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Sofonea, M., Pătrulescu, F.: Penalization of history-dependent variational inequalities. Eur. J. Appl. Math. 25, 155–176 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Zeidler, E.: Nonlinear Functional Analysis and Its Applications. II/B: Nonlinear Monotone Operators. Springer, New York (1990)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Program in Applied Mathematical and Computational Sciences (AMCS), Department of MathematicsUniversity of IowaIowa CityUSA
  2. 2.Laboratoire de Mathématiques et PhysiqueUniversité de Perpignan Via DomitiaPerpignanFrance

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