From solvability and approximation of variational inequalities to solution of nondifferentiable optimization problems in contact mechanics

Abstract

In this paper, we first gather existence results for linear and for pseudo-monotone variational inequalities in reflexive Banach spaces. We discuss the necessity of the involved coerciveness conditions and their relationship. Then, we combine Mosco convergence of convex closed sets with an approximation of pseudo-monotone bifunctions and provide a convergent approximation procedure for pseudo-monotone variational inequalities in reflexive Banach spaces. Since hemivariational inequalities in linear elasticity are pseudo-monotone, our approximation method applies to nonmonotone contact problems. We sketch how regularization of the involved nonsmooth functionals together with finite element approximation lead to an efficient numerical solution method for these nonconvex nondifferentiable optimization problems. To illustrate our theory, we give a numerical example of a 2D linear elastic block under a given nonmonotone contact law.