Geodesic
Computable a posteriori error estimates in the finite element method based on its local conservativity: improvements using local minimization
Ibrahim Cheddadi1 ; Radek Fučík2 ; Mariana I. Prieto3 ; Martin Vohralík4
1 Univ. Grenoble and CNRS, Laboratoire Jean Kuntzmann, 51 rue des Mathématiques, 38400 Saint Martin d'Hères & INRIA Grenoble-Rhône-Alpes, Inovallée, 655 avenue de l'Europe, Montbonnot 38334 Saint Ismier Cedex, France;
2 Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, 12000 Prague, Czech Republic;
3 Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Intendente Güiraldes 2160, Ciudad Universitaria, C1428EGA, Argentina;
4 UPMC Univ. Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France & CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France;
ESAIM. Proceedings, Tome 24 (2008), pp. 77-96.

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We investigate in this paper improvements of the a posteriori error estimates in the finite element method discretization of the Poisson equation, introduced in [M. Vohralík, A posteriori error estimation in the conforming finite element method based on its local conservativity and using local minimization, C. R. Math. Acad. Sci. Paris 346 (2008), 687–690] and [M. Vohralík, Guaranteed and fully robust a posteriori error estimates for conforming discretizations of diffusion problems with discontinuous coefficients, submitted]. The estimates presented in these references are guaranteed in the sense that they feature no undetermined constants and fully computable but numerical experiments show that the effectivity index, i.e., the ratio of the estimated and actual error, does not approach the optimal value of one but rather a slightly bigger value. We identify in this paper the reason for this and introduce a possible remedy, which consists in performing a local minimization of the values of the estimators over patches of simplicial submesh elements. We then present a set of numerical experiments showing the improvements achieved and compare our estimators, both theoretically and numerically, with the classical residual ones.
DOI : 10.1051/proc:2008031

Ibrahim Cheddadi 1 ; Radek Fučík 2 ; Mariana I. Prieto 3 ; Martin Vohralík 4

1 Univ. Grenoble and CNRS, Laboratoire Jean Kuntzmann, 51 rue des Mathématiques, 38400 Saint Martin d'Hères & INRIA Grenoble-Rhône-Alpes, Inovallée, 655 avenue de l'Europe, Montbonnot 38334 Saint Ismier Cedex, France;
2 Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, 12000 Prague, Czech Republic;
3 Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Intendente Güiraldes 2160, Ciudad Universitaria, C1428EGA, Argentina;
4 UPMC Univ. Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France & CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France; 
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     author = {Ibrahim Cheddadi and Radek Fu\v{c}{\'\i}k and Mariana I. Prieto and Martin Vohral{\'\i}k},
     title = {Computable a posteriori error estimates in the finite element method based on its local conservativity: improvements using local minimization},
     journal = {ESAIM. Proceedings},
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     doi = {10.1051/proc:2008031},
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Ibrahim Cheddadi; Radek Fučík; Mariana I. Prieto; Martin Vohralík. Computable a posteriori error estimates in the finite element method based on its local conservativity: improvements using local minimization. ESAIM. Proceedings, Tome 24 (2008), pp. 77-96. doi : 10.1051/proc:2008031. http://geodesic.mathdoc.fr/articles/10.1051/proc:2008031/

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