Talks and Poster Presentations (with Proceedings-Entry):

C. Carstensen, M. Feischl, D. Praetorius:
"Rate optimality of adaptive algorithms, part II: Extensions";
Talk: 11th World Congress on Computational Mechanics (WCCM XI), Barcelona; 07-20-2014 - 07-25-2014; in: "11th World Congress on Computational Mechanics (WCCM XI)", (2014), 2511 - 2522.

English abstract:
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 in the spirit of [Stevenson 2007]. For this
purpose, an overall set of four axioms on the error estimator is sufficient and (partially even)

Compared to the state of the art in the temporary literature, the improvements 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

Electronic version of the publication:

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