Variational-hemivariational inequalities: A brief survey on mathematical theory and numerical analysis
Weimin Han
TL;DR
This paper surveys the theory and numerical analysis of variational-hemivariational inequalities (VHI), highlighting their role as a unifying framework for non-smooth and set-valued relations in mechanics. It presents an accessible, non-pseudomonotone based approach to well-posedness for abstract stationary VHIs and develops Galerkin/FEM methods with Céa-type error estimates. The framework is then applied to representative elasticity problems (VE, VI, VHI) and to mixed VHIs in fluid mechanics (Stokes and Navier–Stokes), with convergence results and regularity-based error bounds. The discussion extends to numerical methods (DG, VEM), optimization-based schemes, and time-dependent/history-dependent VHIs, underscoring the practical relevance and future research directions in computational mechanics.
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.