Anisotropic error control for environmental applications

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 [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 introduced 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. Perotto, 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 European 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 Engrg. 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 elements.