Abstract
In this paper we develop a finite element approximationfor vector-valuedhemivariational inequalities.This class of hemivariational problems wasintroducedin [12],[13]. We study two differentproblems: unconstrained oneand constrained one witha nonempty, closed, convex constraint set K.
We shall show firstly that the discrete problemsare solvable by usingconsequences of Kakutanifixed point theorem and secondly that the solutionsof the discrete problemsare close on subsequences to the continuous ones.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Browder F. E. and Hess P. (1972), Nonlinear Mappings of Monotone Type in Banach Spaces, J. Funct. Anal., 11, 251–294.
Ciarlet P. G. (1978), The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam.
Clarke F. H. (1983), Optimization and Nonsmooth Analysis, Wiley Interscience, New York.
Glowinski R., Lions J. L. and Trémolièrs R. (1981), Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam.
Glowinski R. (1984), Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, New York.
Hlaváček I, Haslinger J., Nečas J., and Lovišek J. (1988), Solutions of Variational Inequalitiesin Mechanics, Springer-Verlag, New York.
Miettinen M. and Haslinger J. (1992), Approximation of Optimal Control Problems of Hemivariational Inequalities, Numer. Funct. Anal. and Optimiz. 13(1&2), 43–68.
Miettinen M. and Haslinger J. (1995), Approximation of Nonmonotone Multivalued Differential Inclusions, IMA J. Num. Anal. 15, 475–503.
Miettinen M. (1995), On Constrained Hemivariational Inequalities and Their Approximation, Applicable Analysis 56, 303–327
Miettinen M. (1993), Approximation of Hemivariational Inequalities and Optimal Control Problems, PhD Thesis, University of Jyväskylä, Department of Mathematics, Report 59.
Mäkelä M. M. and Neittanmäki P. (1992), Nonsmooth Optimization. Analysis and Algorithms with Applications to Optimal Control, World Scientific, Singapore.
Naniewicz Z. and Panagiotopoulos P. D. (1995), Mathematical Theory of Hemivariational Inequalities and Applications, Marcel Dekker, New York.
Naniewicz Z., Hemivariational Inequalities with Functions Fulfilling Directional Growth Condition, to appear in Applicable Analysis.
Nečas J. (1967), Les méthodes directes en théorie des équations elliptiques, Academia, Prague.
Panagiotopoulos P. D. (1985), Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions, Birkhäuser, Basel.
Panagiotopoulos P. D. (1991), Coercive and Semicoercive Hemivariational Inequalities, Nonlin. Anal. 16, 209–231.
Panagiotopoulos P. D. (1993) Hemivariational Inequalities. Applications in Mechanics and Engineering, Springer-Verlag, New York.
Rockafellar R. T. (1970), Convex Analysis, Princeton Univ. Press, Princeton.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Miettinen, M., Haslinger, J. Finite Element Approximation of Vector-Valued Hemivariational Problems. Journal of Global Optimization 10, 17–35 (1997). https://doi.org/10.1023/A:1008234502169
Issue Date:
DOI: https://doi.org/10.1023/A:1008234502169