Numerical Analysis
Stability of discrete liftings
Comptes Rendus. Mathématique, Volume 337 (2003) no. 12, pp. 805-808.

In this short Note we prove the equivalence between having a discrete lifting of Dirichlet boundary conditions for (abstract) finite element spaces and having a Scott–Zhang type operator in the space, i.e., a stable projection preserving homogeneous boundary conditions. Both results are equivalent to the possibility of obtaining a Céa estimate where approximation of the boundary conditions is separated from the approximation capabilities of the space.

Dans cette courte Note nous démontrons l'équivalence entre l'existence d'un relèvement discret des conditions aux limites de Dirichlet pour un espace (abstrait) d'éléments finis et l'existence d'un opérateur de Scott–Zhang sur l'espace, c'est-à-dire, d'une projection stable qui préserve les conditions aux limites homogènes. Ces deux résultats sont équivalents à la possibilité d'obtenir une estimation de Céa, où l'approximation des conditions aux limites est séparée des propriétés d'approximation de l'espace.

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Accepted:
Published online:
DOI: 10.1016/j.crma.2003.10.025
Domínguez, Vı́ctor 1; Sayas, Francisco-Javier 2

1 Dep. Matemática e Informática, Univ. Pública de Navarra, Campus de Arrosadı́a, 31006, Pamplona, Spain
2 Dep. Matemática Aplicada, C.P.S., Universidad de Zaragoza, 50018 Zaragoza, Spain
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Domínguez, Vı́ctor; Sayas, Francisco-Javier. Stability of discrete liftings. Comptes Rendus. Mathématique, Volume 337 (2003) no. 12, pp. 805-808. doi : 10.1016/j.crma.2003.10.025. http://www.numdam.org/articles/10.1016/j.crma.2003.10.025/

[1] Brezzi, F.; Fortin, M. Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991

[2] Engl, H.W.; Hanke, M.; Neubauer, A. Regularization of Inverse Problems, Kluwer Academic, Dordrecht, 1996

[3] Girault, V.; Scott, L.R. Hermite interpolation of nonsmooth functions preserving boundary conditions, Math. Comp., Volume 71 (2002), pp. 1043-1074

[4] Scott, L.R.; Zhang, S. Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp., Volume 54 (1990), pp. 483-493

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