Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Robust a-posteriori estimator for advection-diffusion-reaction problems

Author: Giancarlo Sangalli
Journal: Math. Comp. 77 (2008), 41-70
MSC (2000): Primary 65N30, 65G99
Published electronically: May 14, 2007
MathSciNet review: 2353943
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We propose an almost-robust residual-based a-posteriori estimator for the advection-diffusion-reaction model problem.

The theory is developed in the one-dimensional setting. The numerical error is measured with respect to a norm which was introduced by the author in 2005 and somehow plays the role that the energy norm has with respect to symmetric and coercive differential operators. In particular, the mentioned norm possesses features that allow us to obtain a meaningful a-posteriori estimator, robust up to a $ \sqrt{\log(Pe)}$ factor, where $ Pe$ is the global Péclet number of the problem. Various numerical tests are performed in one dimension, to confirm the theoretical results and show that the proposed estimator performs better than the usual one known in literature.

We also consider a possible two-dimensional extension of our result and only present a few basic numerical tests, indicating that the estimator seems to preserve the good features of the one-dimensional setting.

References [Enhancements On Off] (What's this?)

  • 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), no. 2, 331-353. MR 1668250 (99k:65083)
  • 2. M. Ainsworth and J. T. Oden, A posteriori error estimation in finite element analysis, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2000. MR 1885308 (2003b:65001)
  • 3. R. Araya, E. Behrens, and R. Rodríguez, An adaptive stabilized finite element scheme for the advection-reaction-diffusion equation, Appl. Numer. Math. 54 (2005), no. 3-4, 491-503. MR 2149365 (2006a:65152)
  • 4. R. Araya, A. H. Poza, and E. P. Stephan, A hierarchical a posteriori error estimate for an advection-diffusion-reaction problem, Math. Models Methods Appl. Sci. 15 (2005), no. 7, 1119-1139. MR 2151800 (2006m:65235)
  • 5. I. Babuška, A. Miller, and M. Vogelius, Adaptive methods and error estimation for elliptic problems of structural mechanics, Adaptive computational methods for partial differential equations (College Park, Md., 1983), SIAM, Philadelphia, PA, 1983, pp. 57-73. MR 792521 (87c:65133)
  • 6. S. Berrone, Robustness in a posteriori error analysis for FEM flow models, Numer. Math. 91 (2002), no. 3, 389-422. MR 1907865 (2003d:76103)
  • 7. S. Berrone and C. Canuto, Multilevel a posteriori error analysis for reaction-convection-diffusion problems, Appl. Numer. Math. 50 (2004), no. 3-4, 371-394. MR 2074010 (2005g:65166)
  • 8. F. Brezzi and A. Russo, Choosing bubbles for advection-diffusion problems, Math. Models Methods Appl. Sci. 4 (1994), no. 4, 571-587. MR 1291139 (95h:76079)
  • 9. A. N. Brooks and T. J. R. Hughes, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 32 (1982), no. 1-3, 199-259. MR 679322 (83k:76005)
  • 10. G. Hauke, M. H. Doweidar, and M. Miana, The multiscale approach to error estimation and adaptivity, Comput. Methods Appl. Mech. Engrg. 195 (2006), no. 13-16, 1573-1593. MR 2203982 (2006i:76065)
  • 11. P. Houston, R. Rannacher, and E. Süli, A posteriori error analysis for stabilised finite element approximations of transport problems, Comput. Methods Appl. Mech. Engrg. 190 (2000), no. 11-12, 1483-1508. MR 1807010 (2002d:65115)
  • 12. G. Kunert, A posteriori error estimation for convection dominated problems on anisotropic meshes, Math. Methods Appl. Sci. 26 (2003), no. 7, 589-617. MR 1967323 (2004e:65123)
  • 13. J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Springer-Verlag, New York, 1972, Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181.
  • 14. J.-L. Lions and J. Peetre, Sur une classe d'espaces d'interpolation, Inst. Hautes Études Sci. Publ. Math. (1964), no. 19, 5-68. MR 0165343 (29:2627)
  • 15. P. Morin, R. H. Nochetto, and K. G. Siebert, Convergence of adaptive finite element methods, SIAM Rev. 44 (2002), no. 4, 631-658 (2003). MR 1980447
  • 16. G. Rapin and G. Lube, A stabilized scheme for the Lagrange multiplier method for advection-diffusion equations, Math. Models Methods Appl. Sci. 14 (2004), no. 7, 1035-1060. MR 2076484 (2005c:76064)
  • 17. A. Russo, A posteriori error estimators via bubble functions, Math. Models Methods Appl. Sci. 6 (1996), no. 1, 33-41. MR 1373335 (97e:65112)
  • 18. G. Sangalli, A robust a posteriori estimator for the residual-free bubbles method applied to advection-diffusion problems, Numer. Math. 89 (2001), no. 2, 379-399. MR 1855830 (2002f:65167)
  • 19. -, Construction of a natural norm for the convection-diffusion-reaction operator, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 7 (2004), no. 2, 336-355. MR 2072940 (2005e:35042)
  • 20. -, On robust a posteriori estimators for the advection-diffusion-reaction problem, Tech. Report 04-55, ICES Report, 2004.
  • 21. -, A uniform analysis of nonsymmetric and coercive linear operators, SIAM J. Math. Anal. 36 (2005), no. 6, 2033-2048. MR 2178232 (2006g:35037)
  • 22. H. Triebel, Interpolation theory, function spaces, differential operators, second ed., Johann Ambrosius Barth, Heidelberg, 1995. MR 1328645 (96f:46001)
  • 23. R. Verfürth, A posteriori error estimators for convection-diffusion equations, Numer. Math. 80 (1998), no. 4, 641-663. MR 1650051 (99j:65212)
  • 24. -, Robust a posteriori error estimators for a singularly perturbed reaction-diffusion equation, Numer. Math. 78 (1998), no. 3, 479-493. MR 1603287 (99c:65218)
  • 25. -, Robust a posteriori error estimates for stationary convection-diffusion equations, SIAM J. Numer. Anal. 43 (2005), no. 4, 1766-1782. MR 2182149 (2007d:65116)
  • 26. M. Vohralík, A posteriori error estimates for lowest-order mixed finite element discretizations of convection-diffusion-reaction equations, to appear in SIAM J. Numer. Anal.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65N30, 65G99

Retrieve articles in all journals with MSC (2000): 65N30, 65G99

Additional Information

Giancarlo Sangalli
Affiliation: Dipartimento di Matematica, Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy

Received by editor(s): December 6, 2004
Received by editor(s) in revised form: November 29, 2006
Published electronically: May 14, 2007
Additional Notes: The author was supported in part by the PRIN 2004 project of the Italian MIUR
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society