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Numerical analysis of history-dependent variational–hemivariational inequalities with applications to contact problems†

Published online by Cambridge University Press:  20 April 2015

MIRCEA SOFONEA ,
WEIMIN HAN  and
STANISŁAW MIGÓRSKI
MIRCEA SOFONEA
Affiliation:
Laboratoire de Mathématiques et Physique, Université de Perpignan Via Domitia, 52 Avenue Paul Alduy, 66860 Perpignan, France email: sofonea@univ-perp.fr
WEIMIN HAN
Affiliation:
Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA; School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, Shaanxi 710049, China email: weimin-han@uiowa.edu
STANISŁAW MIGÓRSKI
Affiliation:
Faculty of Mathematics and Computer Science, Jagiellonian University, Institute of Computer Science, ul. Stanisława Łojasiewicza 6, 30348 Krakow, Poland email: stanislaw.migorski@gmail.com
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Abstract

A new class of history-dependent variational–hemivariational inequalities was recently studied in Migórski et al. (2015Nonlinear Anal. Ser. B: Real World Appl.22, 604–618). There, an existence and uniqueness result was proved and used in the study of a mathematical model which describes the contact between a viscoelastic body and an obstacle. The aim of this paper is to continue the analysis of the inequalities introduced in Migórski et al. (2015Nonlinear Anal. Ser. B: Real World Appl.22, 604–618) and to provide their numerical analysis. We start with a continuous dependence result. Then we introduce numerical schemes to solve the inequalities and derive error estimates. We apply the results to a quasistatic frictional contact problem in which the material is modelled with a viscoelastic constitutive law, the contact is given in the form of normal compliance, and friction is described with a total slip-dependent version of Coulomb's law.

Keywords

Variational–hemivariational inequalityClarke subdifferentialhistory-dependent operatorContinuous dependenceFinite element methodConvergenceError estimatesViscoelastic materialFrictional contactWeak solution
Type
Papers
Information
European Journal of Applied Mathematics , Volume 26 , Issue 4 , August 2015 , pp. 427 - 452
Copyright
Copyright © Cambridge University Press 2015 

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Footnotes

†

Research supported by the Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme under Grant Agreement No. 295118, the National Science Center of Poland under the Maestro Advanced Project no. DEC-2012/06/A/ST1/00262, and the Polonium Project No. 31155YH/2014 between the University of Perpignan and the Jagiellonian University. The second author is also partially supported by grants from the Simons Foundation. The third author is also partially supported by the International Project co-financed by the Ministry of Science and Higher Education of Republic of Poland under Grant No. W111/7.PR/2012.

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