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

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.

💡 Research Summary

This survey paper provides a comprehensive overview of the mathematical theory and numerical analysis of variational‑hemivariational inequalities (VHIs), a class of problems that extends the classical variational inequalities (VIs) to include non‑monotone, possibly set‑valued relations that arise in many physical and engineering applications. The authors begin with a historical introduction, recalling the Signorini problem as the first variational inequality and describing how the field has evolved to incorporate non‑smooth, non‑convex phenomena through hemivariational inequalities (HVIs). They emphasize that a VHI can be viewed as a unifying framework that contains both VIs (when the non‑convex part is monotone) and pure variational equations as special cases.

Section 2 reviews the essential tools of nonsmooth analysis: Clarke’s generalized directional derivative and generalized sub‑differential for locally Lipschitz functions. The paper introduces the relaxed monotonicity condition (often written with a constant αΨ) that quantifies the degree of non‑convexity of a functional. When αΨ = 0 the functional is convex; for αΨ > 0 the functional satisfies a weakened monotonicity inequality that still permits existence proofs. These concepts are the backbone of the subsequent well‑posedness analysis.

In Section 3 three mechanical model problems are presented to illustrate the hierarchy: a variational equation (linear elasticity with standard boundary conditions), a variational inequality (frictionless contact), and a variational‑hemivariational inequality (contact with friction, adhesion, or other non‑convex interface laws). By comparing the three, the authors make clear how physical non‑smoothness and non‑monotonicity translate into the mathematical structure of the problem.

Section 4 is the theoretical core. The authors consider an abstract stationary VHI of the form: find u∈V such that
 ⟨Au, v−u⟩ + Ψ⁰(u; v−u) ≥ ⟨f, v−u⟩ ∀v∈K,
where A is a monotone, coercive linear operator, Ψ is locally Lipschitz, Ψ⁰ denotes its Clarke directional derivative, and K is a closed convex set. Instead of relying on the classical surjectivity results for pseudomonotone operators, they propose an alternative proof based on splitting the problem into a monotone part (A) and a non‑monotone part (Ψ) and applying a Browder‑Minty type argument combined with Galerkin approximations. This “accessible” approach avoids heavy functional‑analytic machinery and is presented in detail in Subsection 4.1. Subsection 4.2 summarizes the main numerical analysis results for the abstract VHI: existence of finite‑element approximations, convergence under mesh refinement, and optimal order error estimates (first‑order in the H¹‑norm for linear elements).

Section 5 applies the abstract theory to the contact VHI introduced earlier. Under standard regularity assumptions (e.g., the exact solution belongs to H²(Ω)), the authors prove existence, uniqueness, and derive an error estimate of order O(h) for the linear finite‑element method, confirming that the non‑convex interface terms do not degrade the convergence rate.

Section 6 expands the discussion to mixed VHIs, focusing on Stokes and Navier‑Stokes hemivariational inequalities that arise in fluid mechanics with non‑standard slip or friction boundary conditions. The mixed formulation involves both velocity and pressure variables, requiring compatible finite‑element spaces that satisfy the Ladyzhenskaya‑Babuška‑Brezzi (LBB) condition. The paper outlines existence results for these mixed problems and mentions that discontinuous Galerkin and virtual element methods have been successfully applied, offering flexibility in handling complex geometries and non‑conforming meshes.

Section 7 briefly surveys recent developments for non‑stationary (time‑dependent) and history‑dependent VHIs. Time discretization combined with history integrals leads to a sequence of stationary VHIs at each time step; the authors note that similar well‑posedness and error‑analysis techniques extend to this setting. They also point to alternative algorithms such as ADMM, primal‑dual splitting, and augmented Lagrangian methods that have been adapted to solve VHIs efficiently.

Overall, the paper succeeds in presenting a unified, accessible framework for the analysis of variational‑hemivariational inequalities. By grounding the theory in Clarke’s sub‑differential calculus and the relaxed monotonicity condition, the authors provide a clear pathway from abstract existence proofs to concrete finite‑element error estimates. The inclusion of mechanical examples, mixed fluid‑mechanics problems, and a discussion of modern numerical schemes makes the survey valuable both to mathematicians interested in nonsmooth analysis and to engineers seeking robust computational tools for non‑monotone contact, friction, and fluid‑structure interaction problems.