Rothe method and numerical analysis for history-dependent hemivariational inequalities with applications to contact mechanics

In this paper an abstract evolutionary hemivariational inequality with a history-dependent operator is studied. First, a result on its unique solvability and solution regularity is proved by applying the Rothe method. Next, we introduce a numerical scheme to solve the inequality and derive error estimates. We apply the results to a quasistatic frictional contact problem in which the material is modeled with a viscoelastic constitutive law, the contact is given in the form of multivalued normal compliance, and friction is described with a subgradient of a locally Lipschitz potential. Finally, for the contact problem we provide the optimal error estimate

A class of differential hemivariational inequalities in Banach spaces

In this paper we investigate an abstract system which consists of a hemivariational inequality of parabolic type combined with a nonlinear evolution equation in the framework of an evolution triple of spaces which is called a differential hemivariational inequality [(DHVI), for short]. A hybrid iterative system corresponding to (DHVI) is introduced by using a temporally semi-discrete method based on the backward Euler difference scheme, i.e., the Rothe method, and a feedback iterative technique. We apply a surjectivity result for pseudomonotone operators and properties of the Clarke subgradient operator to establish existence and a priori estimates for solutions to an approximate problem. Finally, through a limiting procedure for solutions of the hybrid iterative system, the solvability of (DHVI) is proved without imposing any convexity condition on the nonlinear function u↦f(t,x,u) and compactness of C0-semigroup eA(t)

Convergence of Rothe scheme for hemivariational inequalities of parabolic type

This article presents the convergence analysis of a sequence of piecewise constant and piecewise linear functions obtained by the Rothe method to the solution of the first order evolution partial differential inclusion $u'(t)+Au(t)+\iota^*\partial J(\iota u(t))\ni f(t)$, where the multivalued term is given by the Clarke subdifferential of a locally Lipschitz functional. The method provides the proof of existence of solutions alternative to the ones known in literature and together with any method for underlying elliptic problem, can serve as the effective tool to approximate the solution numerically. Presented approach puts into the unified framework known results for multivalued nonmonotone source term and boundary conditions, and generalizes them to the case where the multivalued term is defined on the arbitrary reflexive Banach space as long as appropriate conditions are satisfied. In addition the results on improved convergence as well as the numerical examples are presented.Comment: to appear in: International Journal of Numerical Analysis and Modelin

Global attractors for multivalued semiflows with weak continuity properties

A method is proposed to deal with some multivalued semiflows with weak continuity properties. An application to the reaction-diffusion problems with nonmonotone multivalued semilinear boundary condition and nonmonotone multivalued semilinear source term is presented.Comment: to appear in Nonlinear Analysis Series A, Theory, Methods & Application

Existence result for differential inclusion with p(x)-Laplacian

In this paper we study the nonlinear elliptic problem with p(x)-Laplacian (hemivariational inequality). We prove the existence of a nontrivial solution. Our approach is based on critical point theory for locally Lipschitz functionals due to Chang

Differential variational-hemivariational inequalities: existence, uniqueness, stability, and convergence

The goal of this paper is to study a comprehensive systemcalled differential variational–hemivariational inequality which is com-posed of a nonlinear evolution equation and a time-dependentvariational–hemivariational inequality in Banach spaces. Under the gen-eral functional framework, a generalized existence theorem for differ-ential variational–hemivariational inequality is established by employ-ing KKM principle, Minty’s technique, theory of multivalued analysis,the properties of Clarke’s subgradient. Furthermore, we explore a well-posedness result for the system, including the existence, uniqueness, andstability of the solution in mild sense. Finally, using penalty methods tothe inequality, we consider a penalized problem-associated differentialvariational–hemivariational inequality, and examine the convergence re-sult that the solution to the original problem can be approached, as aparameter converges to zero, by the solution of the penalized problem