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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Atkinson, K., Han, W.: Theoretical Numerical Analysis: A Functional Analysis Framework, 3rd edn. Springer, New York (2009)
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)
Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)
Chouly, F., Hild, P.: On convergence of the penalty method for unilateral contact problems. Appl. Numer. Math. 65, 27–48 (2013)
Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Theory. Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York (2003)
Eck, C., Jarušek, J.: Existence results for the static contact problem with Coulomb friction. Math. Mod. Methods Appl. 8, 445–463 (1988)
Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976)
Girault, V., Raviart, P.-A.: Finite Element Methods for Navier–Stokes Equations, Theory and Algorithms. Springer, Berlin (1986)
Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984)
Glowinski, R., Lions, J.-L., Trémolières, R.: Numerical Analysis of Variational Inequalities. North-Holland, Amsterdam (1981)
Gwinner, J., Stephan, E.: Advanced Boundary Element Methods: Treatment of Boundary Value, Transmission and Contact Problems. Springer, New York (2018)
Han, W.: Numerical analysis of stationary variational–hemivariational inequalities with applications in contact mechanics. Math. Mech. Solids 23, 279–293 (2018)
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)
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)
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)
Han, W., Sofonea, M., Barboteu, M.: Numerical analysis of elliptic hemivariational inequalities. SIAM J. Numer. Anal. 55, 640–663 (2017)
Han, W., Sofonea, M., Danan, D.: Numerical analysis of stationary variational–hemivariational inequalities. Numer. Math. 139, 563–592 (2018)
Haslinger, J., Miettinen, M., Panagiotopoulos, P.D.: Finite Element Method for Hemivariational Inequalities. Theory, Methods and Applications. Kluwer Academic Publishers, Boston (1999)
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)
Hlaváček, I., Lovíšek, J.: A finite element analysis for the Signorini problem in plane elastostatics. Apl. Mat. 22, 215–227 (1977)
Kikuchi, N., Oden, J.T.: Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia (1988)
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)
Migórski, S., Ochal, A., Sofonea, M.: A class of variational-hemivariational inequalities in reflexive Banach spaces. J. Elast. 127, 151–178 (2017)
Naniewicz, Z., Panagiotopoulos, P.D.: Mathematical Theory of Hemivariational Inequalities and Applications. Marcel Dekker Inc, New York (1995)
Panagiotopoulos, P.D.: Hemivariational Inequalities, Applications in Mechanics and Engineering. Springer, Berlin (1993)
Sofonea, M., Migórski, S.: Variational-Hemivariational Inequalities with Applications. Chapman & Hall, CRC Press, Boca Raton, London (2018)
Sofonea, M., Migórski, S., Han, W.: A penalty method for history-dependent variational–hemivariational inequalities. Comput. Math. Appl. 75, 2561–2573 (2018)
Sofonea, M., Pătrulescu, F.: Penalization of history-dependent variational inequalities. Eur. J. Appl. Math. 25, 155–176 (2014)
Zeidler, E.: Nonlinear Functional Analysis and Its Applications. II/B: Nonlinear Monotone Operators. Springer, New York (1990)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The work was supported by NSF under the Grant DMS-1521684, and the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie Grant Agreement No. 823731 CONMECH.
Rights and permissions
About this article
Cite this article
Han, W., Sofonea, M. Convergence analysis of penalty based numerical methods for constrained inequality problems. Numer. Math. 142, 917–940 (2019). https://doi.org/10.1007/s00211-019-01036-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-019-01036-8