Improve the efficiency estimate for the a posteriori error estimator

Determine whether the efficiency estimate in Theorem 2.4 for the general a posteriori error estimator can be improved, in particular so that the lower-order trace error term can be omitted for suitable Galerkin approximations of variational inequalities of the second kind.

Background

The paper derives a reliable and efficient a posteriori error estimator for the mixed formulation of abstract variational inequalities of the second kind. The reliability bound controls the primal and dual errors by a residual contribution and a consistency functional. The corresponding efficiency estimate, however, contains not only the primal and dual approximation errors but also a lower-order trace term of the form $\|L(\gamma(u-\tilde u))\|_{0,G}$, or, for the idealized frictional problem, a contact-boundary term involving $\|\gamma_C(u-\tilde u_{hp})\|_{0,E}$.

The authors report numerical evidence suggesting that this additional term may not be necessary in at least some Galerkin settings. They prove an improved estimate under the restrictive condition $$(L(\tilde\lambda),L(\gamma(\tilde u)))_{0,G}=j(\tilde u)$$, but leave unresolved whether a comparable improvement holds more generally or whether the lower-order trace term can be eliminated for suitable approximations. The question is relevant because removing that term would sharpen the efficiency guarantee and more closely align the estimator with the actual discretization error.