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C. Carstensen, M. Feischl, M. Page, D. Praetorius:
"Axioms of adaptivity";
in: "ASC Report 38/2013", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2013, ISBN: 978-3-902627-06-3, 1 - 77.

This paper aims first at a simultaneous axiomatic presentation of the proof of optimal convergence rates for adaptive finite element methods and second at some refinements of particular questions like the avoidance of (discrete) lower bounds, inexact solvers, inhomogeneous boundary data, or the use of equivalent error
estimators. Solely four axioms guarantee the optimality in terms of the error estimators.
Compared to the state of the art in the temporary literature, the improvements of this article can be summarized as follows: First, a general framework is presented which covers the existing literature on
optimality of adaptive schemes. The abstract analysis covers linear as well as nonlinear problems and is independent of the underlying finite element or boundary element method. Second, efficiency of the
error estimator is neither needed to prove convergence nor quasi-optimal convergence behavior of the error estimator. In this paper, efficiency exclusively characterizes the approximation classes involved in terms of the best-approximation error and data resolution and so the the upper bound on the optimal marking
parameters does not depend on the efficiency constant. 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 due to Stevenson 2007. Finally, the general analysis allows for equivalent error estimators and inexact solvers as well as different non-homogeneous and mixed boundary conditions.

finite element method, boundary element method, a posteriori error estimators, adaptive algorithm, local mesh-refinement, convergence, optimality, iterative solvers 1

http://www.asc.tuwien.ac.at/preprint/2013/asc38x2013.pdf