Convergence of Rothe scheme for a class of dynamic variational inequalities involving Clarke subdifferential. Issue 13 (3rd October 2018)

• Record Type: Journal Article
• Authors: Bartosz, Krzysztof
• Is Part Of: Applicable analysis. Volume 97:Issue 13(2018)
• Page Start: 2189
• Page End: 2209
• Publication Date: 2018-10-03
• Subjects: Variational–hemivariational inequality -- Clarke subdifferential -- Rothe method -- non-clamped contact problem -- normal damped response -- unilateral constraint; 49J27 -- 49J45 -- 49J52 -- 49J53 -- 74H15 -- 74H20 -- 74H25 -- 74H30 -- 74S99 -- 74M15; Mathematical analysis -- Periodicals; 515
• DOI: 10.1080/00036811.2017.1359562
• Languages: English
• ISSNs: 0003-6811

Abstract

ABSTRACT: In the first part of the paper we deal with a second-order evolution variational inequality involving a multivalued term generated by a Clarke subdifferential of a locally Lipschitz potential. For this problem we construct a time-semidiscrete approximation, known as the Rothe scheme. We study a sequence of solutions of the semidiscrete approximate problems and provide its weak convergence to a limit element that is a solution of the original problem. Next, we show that the solution is unique and the convergence is strong. In the second part of the paper, we consider a dynamic visco-elastic problem of contact mechanics. We assume that the contact process is governed by a normal damped response condition with a unilateral constraint and the body is non-clamped. The mechanical problem in its weak formulation reduces to a variational–hemivariational inequality that can be solved by finding a solution of a corresponding abstract problem related to one studied in the first part of the paper. Hence, we apply obtained existence result to provide the weak solvability of contact problem.