Navegar

Colección

Datos Estadísticas

Artículo

Este artículo es de Acceso Abierto y puede ser descargado en su versión final desde nuestro repositorio
Consulte el artículo en la página del editor
Consulte la política de Acceso Abierto del editor

Abstract:

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. © 2009 EDP Sciences SMAI.

Registro:

Documento: Artículo
Título:Guaranteed and robust a posteriori error estimates for singularly perturbed reaction-diffusion problems
Autor:Cheddadi, I.; Fučík, R.; Prieto, M.I.; Vohralík, M.
Filiación:Univ. Grenoble, CNRS, Laboratoire Jean Kuntzmann, 51 rue des Mathématiques, 38400 Saint Martin d'Hères, France
INRIA Grenoble-Rhône-Alpes, Inovallée, 655 avenue de l'Europe, Montbonnot, 38334 Saint Ismier Cedex, France
Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, 12000 Prague, Czech Republic
Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Intendente Güiraldes 2160, Ciudad Universitaria, C1428EGA, Argentina
UPMC Univ. Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France
CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France
Palabras clave:A posteriori error estimates; Guaranteed upper bound; Robustness; Singularly perturbed reaction-diffusion problem; Vertex-centered finite volume/finite volume element/box method
Año:2009
Volumen:43
Número:5
Página de inicio:867
Página de fin:888
DOI: http://dx.doi.org/10.1051/m2an/2009012
Título revista:Mathematical Modelling and Numerical Analysis
Título revista abreviado:Math. Model. Numer. Anal.
ISSN:0764583X
PDF:https://bibliotecadigital.exactas.uba.ar/download/paper/paper_0764583X_v43_n5_p867_Cheddadi.pdf
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0764583X_v43_n5_p867_Cheddadi

Referencias:

  • Ainsworth, M., Babǔska, I., Reliable and robust a posteriori error estimating for singularly perturbed reaction-diffusion problems (1999) SIAM J. Numer. Anal, 36, pp. 331-353
  • Bank, R.E., Rose, D.J., Some error estimates for the box method (1987) SIAM J. Numer. Anal, 24, pp. 777-787
  • Bebendorf, M., A note on the Poincaré inequality for convex domains (2003) Z. Anal. Anwendungen, 22, pp. 751-756
  • Brezzi, F., Fortin, M., (1991) Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, 15. , Springer- Verlag, New York
  • Carstensen, C., Funken, S.A., Constants in Clément-interpolation error and residual based a posteriori error estimates in finite element methods (2000) East-West J. Numer. Math, 8, pp. 153-175
  • Cheddadi, I., Fǔcík, R., Prieto, M.I., Vohralík, M., Computable a posteriori error estimates in the finite element method based on its local conservativity: Improvements using local minimization (2008) ESAIM: Proc, 24, pp. 77-96
  • Ern, A., Stephansen, A.F., Vohralík, M., Guaranteed and robust discontinuous Galerkin a posteriori error estimates for convection-diffusion- reaction problems. HAL Preprint 00193540 (2008), submitted for publication; Eymard, R., Galloüet, T., Herbin, R., Finite volume methods (2000) Handbook of Numerical Analysis, 7, pp. 713-1020. , North-Holland, Amsterdam
  • Grosman, S., An equilibrated residual method with a computable error approximation for a singularly perturbed reaction- diffusion problem on anisotropic finite element meshes (2006) ESAIM: M2AN, 40, pp. 239-267
  • Hecht, F., Pironneau, O., Le Hyaric, A., Ohtsuka, K., FreeFem, , http://www.freefem.org, Technical report, Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris, France, /ff, 2007
  • Korotov, S., Two-sided a posteriori error estimates for linear elliptic problems with mixed boundary conditions (2007) Appl. Math, 52, pp. 235-249
  • Kunert, G., Robust a posteriori error estimation for a singularly perturbed reaction-diffusion equation on anisotropic tetrahedral meshes (2001) Adv. Comput. Math, 15, pp. 237-259
  • Luce, R., Wohlmuth, B.I., A local a posteriori error estimator based on equilibrated fluxes (2004) SIAM J. Numer. Anal, 42, pp. 1394-1414
  • Mer, K., Variational analysis of a mixed finite element/finite volume scheme on general triangulations (1994), Technical report, INRIA 2213, France; Payne, L.E., Weinberger, H.F., An optimal Poincaré inequality for convex domains (1960) Arch. Rational Mech. Anal, 5, pp. 286-292
  • Prager, W., Synge, J.L., Approximations in elasticity based on the concept of function space (1947) Quart. Appl. Math, 5, pp. 241-269
  • Repin, S., Sauter, S., Functional a posteriori estimates for the reaction-diffusion problem (2006) C. R. Math. Acad. Sci. Paris, 343, pp. 349-354
  • Roberts, J.E., Thomas, J.-M., Mixed and hybrid methods (1991) Handbook of Numerical Analysis, 2, pp. 523-639. , North-Holland, Amsterdam
  • Stephansen, A.F., (2007) Méthodes de Galerkine discontinues et analyse dérreur a posteriori pour les problémes de diffusion hétérogéne, , Ph.D. Thesis, Ecole nationale des ponts et chaussées, France
  • Vejchodský, T., Guaranteed and locally computable a posteriori error estimate (2006) IMA J. Numer. Anal, 26, pp. 525-540
  • Verf̈urth, R., Robust a posteriori error estimators for a singularly perturbed reaction-diffusion equation (1998) Numer. Math, 78, pp. 479-493
  • Verf̈urth, R., A note on constant-free a posteriori error estimates (2008), Technical report, Ruhr-Universiẗat Bochum, Germany; Vohralík, M., On the discrete Poincaré-Friedrichs inequalities for nonconforming approximations of the Sobolev space H1 (2005) Numer. Funct. Anal. Optim, 26, pp. 925-952
  • Vohralík, M., A posteriori error estimates for lowest-order mixed finite element discretizations of convection-diffusion-reaction equations (2007) SIAM J. Numer. Anal, 45, pp. 1570-1599
  • Vohralík, M., Guaranteed and fully robust a posteriori error estimates for conforming discretizations of diffusion problems with discontinuous coefficients. Preprint R08009, Laboratoire Jacques-Louis Lions (2008), submitted for publication; Vohralík, M., Residual flux-based a posteriori error estimates for finite volume and related locally conservative methods (2008) Numer. Math, 111, pp. 121-158
  • Vohralík, M., Two types of guaranteed (and robust) a posteriori estimates for finite volume methods (2008) Finite Volumes for Complex Applications, 5. , ISTE and John Wiley & Sons, London, UK and Hoboken, USA () 649-656
  • Zienkiewicz, O.C., Zhu, J.Z., A simple error estimator and adaptive procedure for practical engineering analysis (1987) Internat. J. Numer. Methods Engrg, 24, pp. 337-357

Citas:

---------- APA ----------
Cheddadi, I., Fučík, R., Prieto, M.I. & Vohralík, M. (2009) . Guaranteed and robust a posteriori error estimates for singularly perturbed reaction-diffusion problems. Mathematical Modelling and Numerical Analysis, 43(5), 867-888.
http://dx.doi.org/10.1051/m2an/2009012
---------- CHICAGO ----------
Cheddadi, I., Fučík, R., Prieto, M.I., Vohralík, M. "Guaranteed and robust a posteriori error estimates for singularly perturbed reaction-diffusion problems" . Mathematical Modelling and Numerical Analysis 43, no. 5 (2009) : 867-888.
http://dx.doi.org/10.1051/m2an/2009012
---------- MLA ----------
Cheddadi, I., Fučík, R., Prieto, M.I., Vohralík, M. "Guaranteed and robust a posteriori error estimates for singularly perturbed reaction-diffusion problems" . Mathematical Modelling and Numerical Analysis, vol. 43, no. 5, 2009, pp. 867-888.
http://dx.doi.org/10.1051/m2an/2009012
---------- VANCOUVER ----------
Cheddadi, I., Fučík, R., Prieto, M.I., Vohralík, M. Guaranteed and robust a posteriori error estimates for singularly perturbed reaction-diffusion problems. Math. Model. Numer. Anal. 2009;43(5):867-888.
http://dx.doi.org/10.1051/m2an/2009012