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Quasi-Optimal Convergence Rate for an Adaptive Finite Element Method

By J. Manuel Cascon, Christian Kreuzer, Ricardo H. Nochetto and Kunibert G. Siebert


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

Topics: Finite-Elemente-Methode, Konvergenz, Anpassung <Mathematik>, Optimum, ddc:510
Year: 2007
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