Convergence of Rothe scheme for a class of dynamic variational inequalities involving Clarke subdifferential

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.

KEYWORDS:

• Variational–hemivariational inequality
• Clarke subdifferential
• Rothe method
• non-clamped contact problem
• normal damped response
• unilateral constraint

AMS SUBJECT CLASSIFICATIONS:

• 49J27
• 49J45
• 49J52
• 49J53
• 74H15
• 74H20
• 74H25
• 74H30
• 74S99
• 74M15

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

The paper is dedicated to Professor Mircea Sofonea on the occasion of his 60th birthday.

Additional information

Funding

This work was supported by the National Science Center of Poland under the Maestro Advanced Project number DEC-2012/06/A/ST1/00262.