Adaptive finite element methods for elliptic problems : abstract framework and applications
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 3, p. 485-508

We consider a general abstract framework of a continuous elliptic problem set on a Hilbert space V that is approximated by a family of (discrete) problems set on a finite-dimensional space of finite dimension not necessarily included into V. We give a series of realistic conditions on an error estimator that allows to conclude that the marking strategy of bulk type leads to the geometric convergence of the adaptive algorithm. These conditions are then verified for different concrete problems like convection-reaction-diffusion problems approximated by a discontinuous Galerkin method with an estimator of residual type or obtained by equilibrated fluxes. Numerical tests that confirm the geometric convergence are presented.

DOI : https://doi.org/10.1051/m2an/2010010
Classification:  65N30,  65N15,  65N50
Keywords: a posteriori estimator, adaptive FEM, discontinuous Galerkin FEM
@article{M2AN_2010__44_3_485_0,
     author = {Nicaise, Serge and Cochez-Dhondt, Sarah},
     title = {Adaptive finite element methods for elliptic problems : abstract framework and applications},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {44},
     number = {3},
     year = {2010},
     pages = {485-508},
     doi = {10.1051/m2an/2010010},
     zbl = {1191.65158},
     mrnumber = {2666652},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2010__44_3_485_0}
}
Nicaise, Serge; Cochez-Dhondt, Sarah. Adaptive finite element methods for elliptic problems : abstract framework and applications. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 3, pp. 485-508. doi : 10.1051/m2an/2010010. http://www.numdam.org/item/M2AN_2010__44_3_485_0/

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