Keywords: hemivariational inequalities, nonconvex functions error estimates, finite element, material modelling.

In this paper we consider a class of variational inequalities known as hemivariational inequalities involving nonconvex functions. Hemivariational inequalities have very important and novel applications in structural analysis, material modelling, transportation and etc. In a hemivariatonal inequality formulation the presence of "friction" leads to nonconservative "forces" which gives rise to nondifferentiable and nonconvex forms. In most cases the issue of the existence of solutions to such forms is an open problem. However, in recent years numerical techniques have been applied to address this problem. Error estimates for various types of variational inequalities involving second order linear and nonlinear elliptic operators have been derived by many authors under sufficient regular solutions. To the best of our knowledge, finite element method has not been considered for the hemivariational inequalities. Hemivariational inequalities are much more complicated due to the presence of the nonlinear terms involving the nonlinear nonlinear terms. This represents a major difficulty in obtaining the error estimates for the finite element approximation of variational inequalities involving the nonlinear terms. In this paper, we consider the finite element approximation of the hemivariational inequalities and derive the error estimate which is of order h in the energy norm. Our result represent a substantial generalization and improvement of the error analysis of finite element approximation of hemivariational inequalities.

To convey the main idea we consider the problem of finding such that

where is a closed convex set in a Hilbert space . Here denotes the generalized directional derivative of the function at in the direction . For the regularity of the solution , we assume the following hypothesis.

We derive the errors estimate for the finite element approximation of the hemivariational inequalities. For this purpose, we first consider a finite dimensional subspace of continuous piecewise linear functions on the triangulation of the polygonal domain vanishing on its boundary Let be the interpolant of such that agrees at all the vertices of the triangulation. For our purpose, it is enough to chose the finite dimensional convex subset only on the vertices of the triangulation }. Thus the finite element approximation of is: Find such that

The error estimate by the following theorem:

Theorem Let the operator be strongly monotone and Lipschitz continuous and the nonlinear function be locally Lipschitz continuous function. If and then

where and are solutions of (1) and (2) respectively and the hypothesis (A) holds.

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