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C. Carstensen, M. Feischl, D. Praetorius:
"Rate optimality of adaptive algorithms: An axiomatic approach";
in: "ASC Report 11/2014", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2014, ISBN: 978-3-902627-07-0, 1 - 12.

Adaptive mesh-refining algorithms dominate the numerical simulations in computational sciences and engineering, because they promise optimal convergence rates in an overwhelming numerical
evidence. The mathematical foundation of optimal convergence rates has recently been completed and shall be discussed in this talk. We aim at a simultaneous axiomatic presentation of the proof
of optimal convergence rates for adaptive finite elements as well as boundary elements. For this purpose, an overall set of four axioms on the error estimator is sufficient and (partially even) necessary.

Compared to the state of the art in the temporary literature, the improvements of our work can be summarized as follows: First, a general framework is presented which covers the existing
literature on rate optimality of adaptive schemes for both, linear as well as nonlinear problems, which is fairly independent of the underlying (conforming, nonconforming, or mixed) finite element
or boundary element method. Second, efficiency of the error estimator is not needed. Instead, efficiency exclusively characterizes the approximation classes involved in terms of the bestapproximation error plus data resolution. Third, some general quasi-Galerkin orthogonality is not only sufficient, but also necessary for the R-linear convergence of the error estimator, which is
a fundamental ingredient in the current quasi-optimality analysis. Finally, the general analysis allows for various generalizations like equivalent error estimators and inexact solvers as well as different non-homogeneous and mixed boundary conditions.

ascreport, p21732, npde

http://www.asc.tuwien.ac.at/preprint/2014/asc11x2014.pdf