Guaranteed and robust a posteriori error estimates for singularly perturbed reaction-diffusion problems

Cheddadi, Ibrahim; Fučík, Radek; Prieto, Mariana I.; Vohralík, Martin

ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 5, pp. 867-888.

We derive a posteriori error estimates for singularly perturbed reaction-diffusion problems which yield a guaranteed upper bound on the discretization error and are fully and easily computable. Moreover, they are also locally efficient and robust in the sense that they represent local lower bounds for the actual error, up to a generic constant independent in particular of the reaction coefficient. We present our results in the framework of the vertex-centered finite volume method but their nature is general for any conforming method, like the piecewise linear finite element one. Our estimates are based on a H(div)-conforming reconstruction of the diffusive flux in the lowest-order Raviart-Thomas-Nédélec space linked with mesh dual to the original simplicial one, previously introduced by the last author in the pure diffusion case. They also rely on elaborated Poincaré, Friedrichs, and trace inequalities-based auxiliary estimates designed to cope optimally with the reaction dominance. In order to bring down the ratio of the estimated and actual overall energy error as close as possible to the optimal value of one, independently of the size of the reaction coefficient, we finally develop the ideas of local minimizations of the estimators by local modifications of the reconstructed diffusive flux. The numerical experiments presented confirm the guaranteed upper bound, robustness, and excellent efficiency of the derived estimates.

Classification : 65N15, 65N30, 76S05
Mots clés : vertex-centered finite volume/finite volume element/box method, singularly perturbed reaction-diffusion problem, a posteriori error estimates, guaranteed upper bound, robustness

[1] M. Ainsworth and I. Babuška, Reliable and robust a posteriori error estimating for singularly perturbed reaction-diffusion problems. SIAM J. Numer. Anal. 36 (1999) 331-353. | MR | Zbl

[2] R.E. Bank and D.J. Rose, Some error estimates for the box method. SIAM J. Numer. Anal. 24 (1987) 777-787. | MR | Zbl

[3] M. Bebendorf, A note on the Poincaré inequality for convex domains. Z. Anal. Anwendungen 22 (2003) 751-756. | MR | Zbl

[4] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics 15. Springer-Verlag, New York (1991). | MR | Zbl

[5] C. Carstensen and S.A. Funken, Constants in Clément-interpolation error and residual based a posteriori error estimates in finite element methods. East-West J. Numer. Math. 8 (2000) 153-175. | MR | Zbl

[6] I. Cheddadi, R. Fučík, M.I. Prieto and M. Vohralík, Computable a posteriori error estimates in the finite element method based on its local conservativity: improvements using local minimization. ESAIM: Proc. 24 (2008) 77-96. | MR | Zbl

[7] A. Ern, A.F. Stephansen and M. Vohralík, Guaranteed and robust discontinuous Galerkin a posteriori error estimates for convection-diffusion-reaction problems. HAL Preprint 00193540, submitted for publication (2008).

[8] R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in Handbook of Numerical Analysis, Vol. VII, North-Holland, Amsterdam (2000) 713-1020. | MR | Zbl

[9] S. Grosman, An equilibrated residual method with a computable error approximation for a singularly perturbed reaction-diffusion problem on anisotropic finite element meshes. ESAIM: M2AN 40 (2006) 239-267. | Numdam | MR | Zbl

[10] F. Hecht, O. Pironneau, A. Le Hyaric and K. Ohtsuka, FreeFem++. Technical report, Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris, France, http://www.freefem.org/ff++ (2007).

[11] S. Korotov, Two-sided a posteriori error estimates for linear elliptic problems with mixed boundary conditions. Appl. Math. 52 (2007) 235-249. | MR | Zbl

[12] G. Kunert, Robust a posteriori error estimation for a singularly perturbed reaction-diffusion equation on anisotropic tetrahedral meshes. Adv. Comput. Math. 15 (2001) 237-259. | MR | Zbl

[13] R. Luce and B.I. Wohlmuth, A local a posteriori error estimator based on equilibrated fluxes. SIAM J. Numer. Anal. 42 (2004) 1394-1414. | MR | Zbl

[14] K. Mer, Variational analysis of a mixed finite element/finite volume scheme on general triangulations. Technical report, INRIA 2213, France (1994).

[15] L.E. Payne and H.F. Weinberger, An optimal Poincaré inequality for convex domains. Arch. Rational Mech. Anal. 5 (1960) 286-292. | MR | Zbl

[16] W. Prager and J.L. Synge, Approximations in elasticity based on the concept of function space. Quart. Appl. Math. 5 (1947) 241-269. | MR | Zbl

[17] S. Repin and S. Sauter, Functional a posteriori estimates for the reaction-diffusion problem. C. R. Math. Acad. Sci. Paris 343 (2006) 349-354. | MR | Zbl

[18] J.E. Roberts and J.-M. Thomas, Mixed and hybrid methods, in Handbook of Numerical Analysis, Vol. II, North-Holland, Amsterdam (1991) 523-639. | MR | Zbl

[19] A.F. Stephansen, Méthodes de Galerkine discontinues et analyse d'erreur a posteriori pour les problèmes de diffusion hétérogène. Ph.D. Thesis, École nationale des ponts et chaussées, France (2007).

[20] T. Vejchodský, Guaranteed and locally computable a posteriori error estimate. IMA J. Numer. Anal. 26 (2006) 525-540. | MR | Zbl

[21] R. Verfürth, Robust a posteriori error estimators for a singularly perturbed reaction-diffusion equation. Numer. Math. 78 (1998) 479-493. | MR | Zbl

[22] R. Verfürth, A note on constant-free a posteriori error estimates. Technical report, Ruhr-Universität Bochum, Germany (2008). | MR

[23] M. Vohralík, On the discrete Poincaré-Friedrichs inequalities for nonconforming approximations of the Sobolev space $H^1$. Numer. Funct. Anal. Optim. 26 (2005) 925-952. | MR | Zbl

[24] M. Vohralík, A posteriori error estimates for lowest-order mixed finite element discretizations of convection-diffusion-reaction equations. SIAM J. Numer. Anal. 45 (2007) 1570-1599. | MR | Zbl

[25] M. Vohralík, Guaranteed and fully robust a posteriori error estimates for conforming discretizations of diffusion problems with discontinuous coefficients. Preprint R08009, Laboratoire Jacques-Louis Lions, submitted for publication (2008).

[26] M. Vohralík, Residual flux-based a posteriori error estimates for finite volume and related locally conservative methods. Numer. Math. 111 (2008) 121-158. | MR | Zbl

[27] M. Vohralík, Two types of guaranteed (and robust) a posteriori estimates for finite volume methods, in Finite Volumes for Complex Applications V, ISTE and John Wiley & Sons, London, UK and Hoboken, USA (2008) 649-656. | MR

[28] O.C. Zienkiewicz and J.Z. Zhu, A simple error estimator and adaptive procedure for practical engineering analysis. Internat. J. Numer. Methods Engrg. 24 (1987) 337-357. | MR | Zbl