Comptes Rendus
Numerical Analysis
A remark on the optimality of adaptive finite element methods
Comptes Rendus. Mathématique, Volume 349 (2011) no. 3-4, pp. 225-228.

Nous démontrons que l'hypothèse de la petitesse du paramètre de marquage θ dans les méthodes d'éléments finis adaptatives peut être évitée dans la démonstration de l'optimalité de l'algorithme. Pour cela, nous introduisons une nouvelle technique basée sur la comparaison de différentes solutions correspondant à des espaces obtenus par différents raffinements d'un maillage donné. On considère des méthodes conformes et non conformes de bas degé sur des maillages en triangles et tetraèdres.

We show that the standard assumption on the smallness of the marking parameter θ in adaptive finite element methods can be avoided for the proof of the optimality of the algorithm. To this end we propose a new technique based on comparison of the solutions of different finite element spaces obtained by different refinements of a given mesh. We consider conforming and nonconforming low-order finite elements on triangular and tetrahedral meshes.

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DOI : 10.1016/j.crma.2010.11.011

Roland Becker 1 ; David Trujillo 1

1 Laboratoire de mathématiques appliquées and INRIA Bordeaux sud-ouest, université de Pau, 64013 Pau cedex, France
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Roland Becker; David Trujillo. A remark on the optimality of adaptive finite element methods. Comptes Rendus. Mathématique, Volume 349 (2011) no. 3-4, pp. 225-228. doi : 10.1016/j.crma.2010.11.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.11.011/

[1] R. Becker; S. Mao An optimally convergent adaptive mixed finite element method, Numer. Math., Volume 111 (2008), pp. 35-54

[2] R. Becker; S. Mao; Z.-C. Shi A convergent adaptive finite element method with optimal complexity, Electron. Trans. Numer. Anal., Volume 30 (2008), pp. 291-304

[3] R. Becker; S. Mao; Z.-C. Shi A convergent nonconforming adaptive finite element method with optimal complexity, SIAM J. Numer. Anal., Volume 47 (2010), pp. 4639-4659

[4] P. Binev; W. Dahmen; R. DeVore Adaptive finite element methods with convergence rates, Numer. Math., Volume 97 (2004), pp. 219-268

[5] D. Braess; R. Hoppe; J. Schöberl A posteriori estimators for obstacle problems by the hypercircle method, Comput. Vis. Sci., Volume 11 (2008), pp. 351-362

[6] J. Cascon; C. Kreuzer; R. Nochetto; K. Siebert Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal., Volume 46 (2008), pp. 2524-2550

[7] M. Crouzeix; P. Raviart Conforming and nonconforming finite element methods for solving the stationary stokes equations, RAIRO Anal. Num., Volume 7 (1973), pp. 33-76

[8] R. DeVore Nonlinear approximation (A. Iserles, ed.), Acta Numerica, vol. 7, Cambridge University Press, 1998, pp. 51-150

[9] R. Stevenson Optimality of a standard adaptive finite element method, Found. Comput. Math., Volume 7 (2007), pp. 245-269

[10] R. Stevenson The completion of locally refined simplicial partitions created by bisection, Math. Comp., Volume 77 (2008), pp. 227-241

[11] R. Verfürth, A note on constant-free a posteriori error estimates, Tech. rep., Ruhr-Universität, Bochum, 2008.

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