Mathematics of Computation

Author(s): Philippe Destuynder; Brigitte Métivet.
Journal: Math. Comp. 68 (1999), 1379-1396.
MSC (1991): Primary 65N30, 65R20, 73C50
Posted: February 24, 1999
Retrieve article in: PDF
This article is available free of charge

Abstract: The goal of this paper is to define a procedure for bounding the error in a conforming finite element method. The new point is that this upper bound is fully explicit and can be computed locally. Numerical tests prove the efficiency of the method. It is presented here for the case of the Poisson equation and a first order finite element approximation.

1.: M. Ainsworth and J. T. Oden [1993], A unified approach to a posteriori error estimation using element residual methods, Numer. Math. 65, 23-50. MR 95a:65185
2.: R. Arcangéli and J. L. Gout [1976], Sur l'évaluation de l'erreur d'interpolation de Lagrange dans un ouvert de ${\mathbb R}^n$ , Rev. Française Automat. Inform. Rech. Opér. Sér. Rouge Anal. Numér. 10, 5-27. MR 58:29586 (preprint)
3.: D. Arnold, J. Douglas, and C. P. Gupta [1984], A family of higher order mixed finite element methods for plane elasticity, Numer. Math. 45, 1-22. MR 86a:65112
4.: I. Babu\v{s}ka and W. C. Rheinboldt [1978], Error estimates for adaptive finite element computation, SIAM J. Numer. Anal. 15, 736-754. MR 58:3400
5.: I. Babu\v{s}ka and R. Rodriguez [1993], The problem of the selection of an a posteriori error indicator based on smoothing techniques, Internat. Numer. Methods Engrg. 36, 539-567. MR 93k:65089
6.: R. E. Bank and A. Weiser [1985], Some a posteriori error estimators for elliptic partial differential equations, Math. Comp. 44, 283-301. MR 86g:65207
7.: C. Bernardi, B Métivet and R. Verfürth [1993], Analyse numérique d'indicateurs d'erreur, note EDF-M172/93062.
8.: P. G. Ciarlet [1978], The finite element method for elliptic problems, North-Holland, Amsterdam. MR 58:25001
9.: M. Collot [1998], Analysis and implementation of explicit error bounds for finite element methods; Static and dynamic aspects, CNAM, Paris.
10.: Ph. Destuynder and B. Métivet [1998], Explicit error bounds for a non-conforming finite element method, Finite Element Methods (Jyväskylä, 1997), Lecture Notes Pure Appl. Math., vol. 196, Marcel Dekker, New York, 1998, pp. 95-111. CMP 98:08
11.: - [1996a], Estimation de l'erreur explicite pour une méthode d'éléments finis non conformes, C. R. Acad. Sci. Paris Sér. I. Math. 322, 1081-1086. MR 97e:65110
12.: - [1996b], Estimation de l'erreur explicite pour une méthode d'éléments finis conformes, C. R. Acad. Sci. Paris Sér. I. Math. 323, 679-684. MR 97h:65138
13.: Ph. Destuynder, M. Collot, and M. Salaün [1997], A mathematical framework for the P. Ladevèze a posteriori error bounds in finite element methods, New Advances in Adaptive Computational Methods in Mechanics (T. Oden and P. Ladevèze, eds.), Elsevier, Amsterdam. (to appear)
14.: K. Eriksson and C. Johnson [1988], An adaptive finite element method for linear elliptic problems, Math. Comp. 50 361-383. MR 89c:65119
15.: L. Gallimard, P. Ladeveze, and J. P. Pelle [1995], La méthode des erreurs en relation de comportement appliquée au contrôle des calculs non linéaires. École CEA EDF-INRIA. Cours du 18-21 septembre 1995 sur le calcul d'erreur a posteriori et adaptation de maillage.
16.: V. Girault and P. A. Raviart [1986], Finite element methods for Navier-Stokes equations. Theory and algorithms, Springer-Verlag, Berlin. MR 88b:65129
17.: P. Ladevèze [1975], Comparaison de modèles de mécanique des milieux continus. Thèse d'état, Université Paris VI, Paris.
18.: P. Ladevèze and D. Leguillon [1983], Error estimate procedure in the finite element method and applications, SIAM J. Numer. Anal. 20, 485-509. MR 84g:65150
19.: J. T. Oden, L. Demkowicz, W. Rachowicz, and T. A. Westermann [1989], Toward a universal $h$ - $p$ adaptive finite element strategy. Part 2. A posteriori error estimation. Comp. Methods Appl. Mech. Engrg. 77, 113-180. MR 91b:65123b
20.: W. Prager and J. L. Synge [1947], Approximations in elasticity based on the concept of function space, Quart. Appl. Math. 5, 241-269. MR 10:81b
21.: P. A. Raviart and J. M. Thomas [1975], A mixed finite element method for second order elliptic problems, Mathematical Aspects of Finite Element Methods (Proc. Conf., C.N.R., Rome, 1975) Lecture Notes in Math., vol. 60b, Springer-Verlag, Berlin, 1977, pp. 292-315. MR 58:3547
22.: - [1983], Introduction à l'analyse numérique des équations aux dérivées partielles, Masson, Paris. MR 87a:65001a
23.: J. E. Roberts and J. M. Thomas [1987], Mixed and hybrid finite element methods, Rapport INRIA n $^\circ$ 737.
24.: A. H. Schatz, L. H. Sloan, and L. B. Wahlbin [1996], Superconvergence in finite element methods and meshes that are locally symmetric with respect to a point, SIAM J. Numer. Anal. 33, 505-521. MR 98f:65112
25.: R. Verfürth [1994], A posteriori error estimation and adaptive mesh-refinement techniques, J. Comput. Appl. Math. 50, 67-83. MR 95c:65171
26.: O. C. Zienkiewicz and J. Z. Zhu [1987], A simple error estimator and adaptive procedure for practical engineering analysis, Internat. J. Numer. Methods Engrg. 24, 337-357. MR 87m:73055
27.: - [1992], The super convergent patch recovery and a posteriori error estimates. Part 1: The recovery technique, Internat. J. Numer. Methods Engrg. 33 1331-1364. MR 93c:73098

Retrieve articles in Mathematics of Computation with MSC (1991): 65N30, 65R20, 73C50

Philippe Destuynder
Affiliation: CNAM/IAT, 15 rue Marat, 78210 Saint-Cyr-L'École, France
Email: destuynd@cnam.fr

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6842(e) ISSN 0025-5718(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS