|Abstract:|| We provide a unifying framework that generalizes the 2D and 3D settings proposed in  and , respectively. In these two works we propose
a gradient recovery type a posteriori error estimator for finite element approximations
on anisotropic meshes. The novelty is the inclusion of the geometrical features of the computational mesh (size, shape and orientation)
in the estimator itself. Moreover, we preserve the good properties of recovery based error estimators, in particular their computational cheapness and ease of implementation. A metric-based optimization procedure, relying on the estimator, drives the anisotropic adaptation of the mesh. The focus of this work then moves to a goal-oriented framework. In particular,
we extend the idea proposed in [32, 17] to the control of a goal functional.
The preliminary results are promising, since it is shown numerically to yield quasi-optimal triangulations with respect to the error-vs-number of elements behaviour.|