Skip to main content
SpringerLink
Log in

Numerical analysis of history-dependent variational-hemivariational inequalities

  • Articles
  • Progress of Projects Supported by NSFC
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

In this paper, numerical analysis is carried out for a class of history-dependent variational-hemivariational inequalities by arising in contact problems. Three different numerical treatments for temporal discretization are proposed to approximate the continuous model. Fixed-point iteration algorithms are employed to implement the implicit scheme and the convergence is proved with a convergence rate independent of the time step-size and mesh grid-size. A special temporal discretization is introduced for the history-dependent operator, leading to numerical schemes for which the unique solvability and error bounds for the temporally discrete systems can be proved without any restriction on the time step-size. As for spatial approximation, the finite element method is applied and an optimal order error estimate for the linear element solutions is provided under appropriate regularity assumptions. Numerical examples are presented to illustrate the theoretical results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adams R A, Fournier J J F. Sobolev Spaces, 2nd ed. Singapore: Academic Press, 2003

    MATH  Google Scholar 

  2. Barboteu M, Bartosz K, Han W, et al. Numerical analysis of a hyperbolic hemivariational inequality arising in dynamic contact. SIAM J Numer Anal,, 2015, 53: 527–550

    Article  MathSciNet  Google Scholar 

  3. Brezis H. Equations et inequations non lineaires dans les espaces vectoriels en dualite. Ann Inst Fourier (Grenoble),, 1968, 18: 115–175

    Article  MathSciNet  Google Scholar 

  4. Carl S, Le V K, Motreanu D. Nonsmooth Variational Problems and Their Inequalities: Comparison Principles and Applications. New York: Springer, 2007

    Book  Google Scholar 

  5. Clarke F H. Generalized gradients and applications. Trans Amer Math Soc,, 1975, 205: 247–262

    Article  MathSciNet  Google Scholar 

  6. Clarke F H. Optimization and Nonsmooth Analysis. New York: Wiley-Interscience, 1983

    MATH  Google Scholar 

  7. Glowinski R. Numerical Methods for Nonlinear Variational Problems. New York: Springer, 1984

    Book  Google Scholar 

  8. Glowinski R, Lions J L, Trémolières R. Numerical Analysis of Variational Inequalities. Amsterdam: North-Holland, 1981

    MATH  Google Scholar 

  9. Han W. Numerical analysis of stationary variational-hemivariational inequalities with applications in contact mechanics. Math Mech Solids,, 2018, 23: 279–293

    Article  MathSciNet  Google Scholar 

  10. Han W, Migórski S, Sofonea M. A class of variational-hemivariational inequalities with applications to frictional contact problems. SIAM J Math Anal,, 2014, 46: 3891–3912

    Article  MathSciNet  Google Scholar 

  11. Han W, Sofonea M. Numerical analysis of hemivariational inequalities in contact mechanics. Acta Numer,, 2019, 28: 175–286

    Article  MathSciNet  Google Scholar 

  12. Han W, Sofonea M, Barboteu M. Numerical analysis of elliptic hemivariational inequalities. SIAM J Numer Anal,, 2017, 55: 640–663

    Article  MathSciNet  Google Scholar 

  13. Han W, Sofonea M, Danan D. Numerical analysis of stationary variational-hemivariational inequalities. Numer Math,, 2018, 139: 563–592

    Article  MathSciNet  Google Scholar 

  14. Haslinger J, Miettinen M, Panagiotopoulos P D. Finite Element Method for Hemivariational Inequalities. Theory, Methods and Applications. Boston: Kluwer, 1999

    MATH  Google Scholar 

  15. Hlavacek I, Haslinger J, Necas J, et al. Solution of Variational Inequalities in Mechanics. New York: Springer-Verlag, 1988

    Book  Google Scholar 

  16. Kazmi K, Barboteu M, Han W, et al. Numerical analysis of history-dependent quasivariational inequalities with applications in contact mechanics. ESAIM Math Model Numer Anal,, 2014, 48: 919–942

    Article  MathSciNet  Google Scholar 

  17. Kinderlehrer D, Stampacchia G. An Introduction to Variational Inequalities and Their Applications. New York: Academic Press, 1980

    MATH  Google Scholar 

  18. Lions J L, Stampacchia G. Variational inequalities. Comm Pure Appl Math,, 1967, 20: 493–519

    Article  MathSciNet  Google Scholar 

  19. Migórski S, Ochal A, Sofonea M. Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems. Advances in Mechanics and Mathematics, vol. 26. New York: Springer, 2013

    MATH  Google Scholar 

  20. Migórski S, Ochal A, Sofonea M. A class of variational-hemivariational inequalities in reflexive Banach spaces. J Elasticity,, 2017, 127: 151–178

    Article  MathSciNet  Google Scholar 

  21. Naniewicz Z, Panagiotopoulos P D. Mathematical Theory of Hemivariational Inequalities and Applications. New York: Dekker, 1995

    MATH  Google Scholar 

  22. Panagiotopoulos P D. Nonconvex energy functions, hemivariational inequalities and substationary principles. Acta Mech,, 1983, 42: 160–183

    Google Scholar 

  23. Panagiotopoulos P D. Hemivariational Inequalities: Applications in Mechanics and Engineering. Berlin: Springer-Verlag, 1993

    Book  Google Scholar 

  24. Rockafellar R T. Convex Analysis. Princeton: Princeton University Press, 1970

    Book  Google Scholar 

  25. Sofonea M, Matei A. History-dependent quasi-variational inequalities arising in contact mechanics. European J Appl Math,, 2011, 22: 471–491

    Article  MathSciNet  Google Scholar 

  26. Sofonea M, Migórski S. A class of history-dependent variational-hemivariational inequalities. NoDEA Nonlinear Differential Equations Appl,, 2016, 23: 1–23

    Article  MathSciNet  Google Scholar 

  27. Sofonea M, Migórski S. Variational-Hemivariational Inequalities with Applications. Boca Raton-London: Chapman & Hall/CRC Press, 2018

    MATH  Google Scholar 

  28. Xu W, Huang Z, Han W, et al. Numerical analysis of history-dependent variational-hemivariational inequalities with applications in contact mechanics. J Comput Appl Math,, 2019, 351: 364–377

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The fourth author was supported by National Natural Science Foundation of China (Grant Nos. 11671098 and 91630309) and Higher Education Discipline Innovation Project (111 Project) (Grant No. B08018). The fourth author thanks Institute of Scientific Computation and Financial Data Analysis, Shanghai University of Finance and Economics for the support during his visit.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Shufen Wang & Wenbin Chen

  2. Faculty of Science, Tongji Zhejiang College, Jiaxing, 314051, China

    Wei Xu

  3. Department of Mathematics, University of Iowa, Iowa City, IA, 52242, USA

    Weimin Han

Authors
  1. Shufen Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wei Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Weimin Han
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Wenbin Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wenbin Chen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, S., Xu, W., Han, W. et al. Numerical analysis of history-dependent variational-hemivariational inequalities. Sci. China Math. 63, 2207–2232 (2020). https://doi.org/10.1007/s11425-019-1672-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-019-1672-4

Keywords

MSC (2010)

Search

Navigation