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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