On adaptive computational methods: global norms controlling local errors

In this note, we continue our studies on optimised mesh design for the Finite Element (FE) method 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., [3,7,18]), where error control for the primal problem is established by solving an auxiliary (dual) problem. In this context we blamed (cf. [17,18]) global norm estimates being not that useful in applications. But having a closer look at the DWR-concept, one observes that in fact global error bounds can be employed to establish local error control. We derive rigorous error bounds, especially we control the approximation process of the (unknown) dual solution entering the proposed estimate. Additional, these estimates provide information to optimise the approximation process of the primal and dual problem.