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Variational-Hemivariational Inequalities: A Brief Survey on Mathematical Theory and Numerical Analysis
Journal article

Variational-Hemivariational Inequalities: A Brief Survey on Mathematical Theory and Numerical Analysis

Weimin Han
Approximation Theory and Special Functions, Vol.2(1), pp.1-27
04/08/2026
DOI: 10.65135/atsf.2026.6

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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.

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