|Abstract:|| In this paper we aim at controlling physically meaningful quantities with emphasis on environmental applications. This is carried out by an efficient numerical procedure combining the goal-oriented framework [Acta Numer. 10 (2001) 1] with the anisotropic setting introduced in [Numer. Math. 89 (2001) 641].
A first attempt in this direction has been proposed in [Appl.
Numer. Math. 51 (2004) 511]. Here we improve this analysis by carrying over to the goal-oriented framework the good property of the a posteriori error estimator to depend on the error itself, typical of the anisotropic residual based error analysis presented in [Comput. Methods Appl. Mech. Engrg. 195 (2006) 799; Numerical Mathematics and Advanced Applications - Enumath2001 Springer Verlag Italia (2003) 731]. On the one hand this dependence makes the estimator not immediately computable; nevertheless, after approximating this error via the Zienkiewicz-Zhu gradient recovery procedure [Internat. J. Numer.
Methods Engrg. 24 (1987) 337; Internat. J. Numer. Methods Engrg. 33 (1992) 1331], the resulting estimator is expected to exhibit a higher convergence rate than the one in [Appl. Numer. Math. 51 (2004) 511]. As the broad numerical validation attests, the proposed estimator turns out to be more efficient in terms of d.o.f. s per accuracy or equivalently, more accurate for a fixed number of elements.