On computational methods for variational inequalities

In this note, we focus on optimised mesh design for the Finite Element (FE) method for variational inequalities using global norm estimates for local error control. The strategies are based on the so called dual-weighted-residual (DWR) approach to a posteriori error control for FE-schemes (see, e.g., Rannacher et al. [19, 6, 2]), where error control for the primal problem is established by solving an auxiliary (dual) problem. In this context we blamed (cf. e.g., Rannacher and Suttmeier [18, 19]) global norm estimates being not that useful in applications. But having a closer look at the DWR-concept, one observes that in fact global (energy) error bounds can be employed to establish local error control. Our ideas and techniques are illustrated at the socalled obstacle problem.It turns out, that reliable and efficient energy error control is one main ingredient to establish useful a posteriori error bounds for local quantities. Therefore, in addition, we derive an unified approach to a posteriori error control in the energy norm for elliptic variational inequalities of first kind. Eventually, this framework is applied to Signorinis problem.