Uniqueness of the discrete penalized hemivariational inequality

Determine whether the discrete penalized hemivariational inequality $(S^h_{\alpha})$ has a unique solution, including for the nonconvex logarithmic superpotential used in Example 3.

Background

The discrete problem (Sαh)(S^h_{\alpha}) is the finite-element approximation of the penalized hemivariational inequality, with mesh size h>0h>0 and heat-transfer parameter α>0\alpha>0. Under the paper’s assumptions, existence is established, but uniqueness is not guaranteed. Numerical experiments for Example 3 show two distinct solution branches over an interval of penalization parameters, demonstrating that nonuniqueness can occur at finite α\alpha and preventing a uniform error estimate based on an arbitrarily selected solution.

References

Problem ($S_{\alpha}h$) is not known to have a unique solution, and for Example~3 it is not.

— Finite Element Approximation of a Hemivariational Inequality for Steady-State Heat Conduction: Double-Limit Convergence of Penalization and Discretization  (2609.35538 - Bartman-Szwarc et al., 28 Sep 2026) in Section 4, Numerical Results, paragraph immediately preceding Figure \ref{fig:branches}