Semidiscrete numerical approximation for dynamic hemivariational inequalities with history-dependent operators

Abstract

In this paper, we are concerned with a class of second-order hemivariational inequalities involving history-dependent operators. For the problem, we first derive a semidiscrete scheme by implicit Euler formula and prove its unique solvability. The existence and uniqueness of a solution to the inequality problem is given by Rothe method. As the core part of the paper, we propose a two-step semidiscrete approximation for the problem, provide its unique solvability and obtain its second-order error estimates. The two-step scheme is more accurate than the standard implicit Euler scheme. Finally, we apply the results to a dynamic frictionless contact problem with long memory.

Keywords:

• Hemivariational inequality
• contact problem
• history-dependent operator
• error estimates

Mathematic Subject classifications:

• 65M15
• 65N22
• 74H15

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions. This work was supported by the European Unions Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grand Agreement No. 823731 CONMECH.