Anisotropic error control for environmental applications

In this paper we aim at controlling physically meaningful quantities with emphasis on environmental applications. This is car- ried out by an efficient numerical procedure combining the goal-oriented framework [R. Becker, R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods, Acta Numer. 10 (2001) 1–102] with the anisotropic setting in- troduced in [L. Formaggia, S. Perotto, New anisotropic a priori error estimates, Numer. Math. 89 (2001) 641–667]. A first attempt in this direction has been proposed in [L. Formaggia, S. Micheletti, S. Perotto, Anisotropic mesh adaptation in computational fluid dynamics: application to the advection–diffusion–reaction and the Stokes problems, Appl. Numer. Math. 51 (2004) 511–533]. 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 [G. Maisano, S. Micheletti, S. Per- otto, C.L. Bottasso, On some new recovery based a posteriori error estimators, Comput. Methods Appl. Mech. Engrg. 195 (37–40) (2006) 4794–4815; S. Micheletti, S. Perotto, An anisotropic recovery-based a posteriori error estimator, in: F. Brezzi, A. Buffa, S. Corsaro, A. Murli (Eds.), Numerical Mathematics and Advanced Applications—ENUMATH2001, Proceedings of the 4th Euro- pean International Conference on Numerical Mathematics and Advanced Applications, Springer-Verlag, Italia, 2003, pp. 731–741]. On the one hand this dependence makes the estimator not immediately computable; nevertheless, after approximating this error via the Zienkiewicz–Zhu gradient recovery procedure [O.C. Zienkiewicz, J.Z. Zhu, A simple error estimator and adaptive procedure for practical engineering analysis, Internat. J. Numer. Methods Engrg. 24 (2) (1987) 337–357; O.C. Zienkiewicz, J.Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique, Internat. J. Numer. Methods En- grg. 33 (1992) 1331–1364], the resulting estimator is expected to exhibit a higher convergence rate than the one in [L. Formaggia, S. Micheletti, S. Perotto, Anisotropic mesh adaptation in computational fluid dynamics: application to the advection–diffusion– reaction and the Stokes problems, Appl. Numer. Math. 51 (2004) 511–533]. 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 ele- ments.