Inequality problems in mechanics can be divided into two main categories: that of variational inequalities concerned with convex energy functional (potentials), and that of hemivariational inequalities concerned with nonsmooth and nonconvex energy functionals (superpotentials). Through the formulation of hemivariational inequalities, problems involving nonmonotone, nonsmooth and multivalued constitutive laws, forces, and boundary conditions can be treated successfully. Hemivariational inequalities have been shown to be very useful across a wide variety of subjects, ranging from nonsmooth mechanics, physics, engineering, to economics.
This talk starts with a gentle description of the basic notions and ideas of the theory of hemivariational inequalities. We then present new results on convergence and optimal order error estimates for numerical solutions of elliptic hemivariational inequalities. Numerical examples are shown on the performance of the numerical methods, including numerical convergence orders.