Free access
Issue
ESAIM: M2AN
Volume 43, Number 5, September-October 2009
Page(s) 867 - 888
DOI http://dx.doi.org/10.1051/m2an/2009012
Published online 30 April 2009
  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. [CrossRef] [MathSciNet]
  2. R.E. Bank and D.J. Rose, Some error estimates for the box method. SIAM J. Numer. Anal. 24 (1987) 777–787. [CrossRef] [MathSciNet]
  3. M. Bebendorf, A note on the Poincaré inequality for convex domains. Z. Anal. Anwendungen 22 (2003) 751–756. [CrossRef] [MathSciNet]
  4. F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics 15. Springer-Verlag, New York (1991).
  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. [MathSciNet]
  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. [CrossRef]
  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.
  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. [CrossRef] [EDP Sciences]
  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. [CrossRef] [MathSciNet]
  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. [CrossRef] [MathSciNet]
  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. [CrossRef] [MathSciNet]
  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. [CrossRef] [MathSciNet]
  16. W. Prager and J.L. Synge, Approximations in elasticity based on the concept of function space. Quart. Appl. Math. 5 (1947) 241–269. [MathSciNet]
  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. [CrossRef] [MathSciNet]
  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.
  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. [CrossRef] [MathSciNet]
  21. R. Verfürth, Robust a posteriori error estimators for a singularly perturbed reaction–diffusion equation. Numer. Math. 78 (1998) 479–493. [CrossRef] [MathSciNet]
  22. R. Verfürth, A note on constant-free a posteriori error estimates. Technical report, Ruhr-Universität Bochum, Germany (2008).
  23. M. Vohralík, On the discrete Poincaré–Friedrichs inequalities for nonconforming approximations of the Sobolev space H1. Numer. Funct. Anal. Optim. 26 (2005) 925–952. [CrossRef] [MathSciNet]
  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. [CrossRef] [MathSciNet]
  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. [CrossRef] [MathSciNet]
  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.
  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. [CrossRef] [MathSciNet]

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