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

## TL;DR

This paper develops a numerical approach using the Rothe method to solve complex history-dependent hemivariational inequalities, with applications to contact mechanics involving viscoelastic materials and frictional contact.

## Contribution

It introduces a novel numerical scheme with error estimates for solving history-dependent hemivariational inequalities, including an application to a contact mechanics problem.

## Key findings

- Proved unique solvability and regularity of the inequality.
- Derived optimal error estimates for the numerical scheme.
- Applied the method successfully to a contact mechanics problem.

## Abstract

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.