Quasi-Optimal Convergence Rate for an Adaptive Finite Element Method

  • We analyze the simplest and most standard adaptive finite element method (AFEM), with any polynomial degree, for general second order linear, symmetric elliptic operators. As it is customary in practice, AFEM marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property. We prove that AFEM is a contraction for the sum of energy error and scaled error estimator, between two consecutive adaptive loops. This geometric decay is instrumental to derive optimal cardinality of AFEM. We show that AFEM yields a decay rate of energy error plus oscillation in terms of number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity.

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Author:J. Manuel Cascon, Christian Kreuzer, Ricardo H. Nochetto, Kunibert G. SiebertGND
Frontdoor URLhttps://opus.bibliothek.uni-augsburg.de/opus4/514
Series (Serial Number):Preprints des Instituts für Mathematik der Universität Augsburg (2007-09)
Publishing Institution:Universität Augsburg
Release Date:2007/05/30
Tag:error reduction; convergence; optimal cardinality; adaptive algorithm
GND-Keyword:Finite-Elemente-Methode; Konvergenz; Anpassung <Mathematik>; Optimum
Institutes:Mathematisch-Naturwissenschaftlich-Technische Fakultät
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik