M. Aurada, S. Ferraz-Leite, D. Praetorius:

"Estimator reduction and convergence of adaptive FEM and BEM";

in: "ASC Report 27/2009", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2009, ISBN: 978-3-902627-02-5.

We propose a relaxed notion of convergence of adaptive finite element and

boundary element schemes. Instead of asking for convergence of the error to

zero, we only aim to prove estimator convergence in the sense that the

adaptive algorithm drives the underlying error estimator to zero. We observe

that certain error estimators satisfy an estimator reduction property which

is sufficient for estimator convergence. The elementary analysis is only

based on Doerfler marking and inverse estimates, but not on reliability and

efficiency of the error estimator at hand. In particular, this covers certain

adaptive algorithms in the context of FEM and BEM as well as heuristic

strategies which are often successfully used to steer an adaptive anisotropic

mesh-refinement. Our framework therefore contributes to understand adaptivity

in FEM and BEM in a more general sense and gives a first mathematical

justification for the proposed steering of anisotropic mesh-refinements.

http://www.asc.tuwien.ac.at/preprint/2009/asc27x2009.pdf

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