Robust a-posteriori estimator for advection-diffusion-reaction problems
HTML articles powered by AMS MathViewer
- by Giancarlo Sangalli PDF
- Math. Comp. 77 (2008), 41-70 Request permission
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
- Mark Ainsworth and Ivo 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, DOI 10.1137/S003614299732187X
- Mark Ainsworth and J. Tinsley 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, DOI 10.1002/9781118032824
- Rodolfo Araya, Edwin Behrens, and Rodolfo 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, DOI 10.1016/j.apnum.2004.09.015
- Rodolfo Araya, Abner H. Poza, and Ernst 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, DOI 10.1142/S0218202505000674
- 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
- Stefano Berrone, Robustness in a posteriori error analysis for FEM flow models, Numer. Math. 91 (2002), no. 3, 389–422. MR 1907865, DOI 10.1007/s002110100370
- Stefano Berrone and Claudio Canuto, Multilevel a posteriori error analysis for reaction-convection-diffusion problems, Appl. Numer. Math. 50 (2004), no. 3-4, 371–394. MR 2074010, DOI 10.1016/j.apnum.2004.05.002
- Franco Brezzi and Alessandro Russo, Choosing bubbles for advection-diffusion problems, Math. Models Methods Appl. Sci. 4 (1994), no. 4, 571–587. MR 1291139, DOI 10.1142/S0218202594000327
- Alexander N. Brooks and Thomas 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. FENOMECH ”81, Part I (Stuttgart, 1981). MR 679322, DOI 10.1016/0045-7825(82)90071-8
- Guillermo Hauke, Mohamed H. Doweidar, and Mario Miana, The multiscale approach to error estimation and adaptivity, Comput. Methods Appl. Mech. Engrg. 195 (2006), no. 13-16, 1573–1593. MR 2203982, DOI 10.1016/j.cma.2005.05.029
- Paul Houston, Rolf Rannacher, and Endre 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, DOI 10.1016/S0045-7825(00)00174-2
- Gerd Kunert, A posteriori error estimation for convection dominated problems on anisotropic meshes, Math. Methods Appl. Sci. 26 (2003), no. 7, 589–617. MR 1967323, DOI 10.1002/mma.368
- 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.
- J.-L. Lions and J. Peetre, Sur une classe d’espaces d’interpolation, Inst. Hautes Études Sci. Publ. Math. 19 (1964), 5–68 (French). MR 165343
- Pedro Morin, Ricardo H. Nochetto, and Kunibert G. Siebert, Convergence of adaptive finite element methods, SIAM Rev. 44 (2002), no. 4, 631–658 (2003). Revised reprint of “Data oscillation and convergence of adaptive FEM” [SIAM J. Numer. Anal. 38 (2000), no. 2, 466–488 (electronic); MR1770058 (2001g:65157)]. MR 1980447, DOI 10.1137/S0036144502409093
- Gerd Rapin and Gert 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, DOI 10.1142/S0218202504003532
- Alessandro Russo, A posteriori error estimators via bubble functions, Math. Models Methods Appl. Sci. 6 (1996), no. 1, 33–41. MR 1373335, DOI 10.1142/S0218202596000031
- Giancarlo 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, DOI 10.1007/PL00005471
- Giancarlo Sangalli, 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 (English, with English and Italian summaries). MR 2072940
- —, On robust a posteriori estimators for the advection-diffusion-reaction problem, Tech. Report 04-55, ICES Report, 2004.
- Giancarlo Sangalli, A uniform analysis of nonsymmetric and coercive linear operators, SIAM J. Math. Anal. 36 (2005), no. 6, 2033–2048. MR 2178232, DOI 10.1137/S0036141003434996
- Hans Triebel, Interpolation theory, function spaces, differential operators, 2nd ed., Johann Ambrosius Barth, Heidelberg, 1995. MR 1328645
- R. Verfürth, A posteriori error estimators for convection-diffusion equations, Numer. Math. 80 (1998), no. 4, 641–663. MR 1650051, DOI 10.1007/s002110050381
- R. Verfürth, Robust a posteriori error estimators for a singularly perturbed reaction-diffusion equation, Numer. Math. 78 (1998), no. 3, 479–493. MR 1603287, DOI 10.1007/s002110050322
- R. Verfürth, Robust a posteriori error estimates for stationary convection-diffusion equations, SIAM J. Numer. Anal. 43 (2005), no. 4, 1766–1782. MR 2182149, DOI 10.1137/040604261
- 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.
Additional Information
- Giancarlo Sangalli
- Affiliation: Dipartimento di Matematica, Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy
- MR Author ID: 663454
- Email: giancarlo.sangalli@unipv.it
- 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
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 77 (2008), 41-70
- MSC (2000): Primary 65N30, 65G99
- DOI: https://doi.org/10.1090/S0025-5718-07-02018-2
- MathSciNet review: 2353943