Variational-hemivariational inequalities: A brief survey on mathematical theory and numerical analysis

TL;DR

This survey reviews recent advances in the mathematical theory and numerical analysis of variational-hemivariational inequalities, highlighting their applications in physical sciences and engineering involving non-smooth relations.

Contribution

It provides a concise overview of well-posedness and numerical methods for variational-hemivariational inequalities, including theoretical results and practical examples.

Findings

Existence and uniqueness of solutions are established.

Numerical algorithms for solving these inequalities are discussed.

Applications include fluid mechanics and other engineering problems.

Abstract

Variational-hemivariational inequalities are an area full of interesting and challenging mathematical problems. The area can be viewed as a natural extension of that of variational inequalities. Variational-hemivariational inequalities are valuable for application problems from physical sciences and engineering that involve non-smooth and even set-valued relations, monotone or non-monotone, among physical quantities. In the recent years, there has been substantial growth of research interest in modeling, well-posedness analysis, development of numerical methods and numerical algorithms of variational-hemivariational inequalities. This survey paper is devoted to a brief account of well-posedness and numerical analysis results for variational-hemivariational inequalities. The theoretical results are presented for a family of abstract stationary variational-hemivariational inequalities and the main idea is explained for an accessible proof of existence and uniqueness. To better appreciate the distinguished feature of variational-hemivariational inequalities, for comparison, three mechanical problems are introduced leading to a variational equation, a variational inequality, and a variational-hemivariational inequality, respectively. The paper also comments on mixed variational-hemivariational inequalities, with examples from applications in fluid mechanics, and on results concerning the numerical solution of other types (nonstationary, history dependent) of variational-hemivariational inequalities.