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S. Ferraz-Leite, C. Ortner, D. Praetorius:
"Adaptive boundary element method: Simple error estimators and convergence";
Talk: Oberwolfach Workshop on Analysis of Boundary Element Methods, Oberwolfach (invited); 04-13-2008 - 04-19-2008; in: "Analysis of Boundary Element Methods", EMS Publishing House, Oberwolfach Reports, Volume 5, Issue 2 (2008), ISSN: 1660-8933.

In our presentation, the focus is on a standard adaptive
mesh-refining algorithm for the h-version of the boundary
element method (ABEM). The algorithm is steered by an
h-h/2-based error estimator. Unlike to the finite element
method, the usual energy norms in the context of boundary
element methods are nonlocal in the sense that they cannot
be written as the sum of local contributions. Therefore,
we provide some localization of the energy norm in terms of
a weighted L^2-norm to derive the refinement indicators.
In the spirit of recent works on adaptive finite element
methods (AFEM), we prove the convergence of our adaptive
boundary element method.

To the best of our knowledge, the result seems to be the first
contribution on the convergence of standard ABEM. Numerical
experiments are concerned with the weakly singular integral
equation for the Laplacian in 2D and 3D.

Parts of the results have been achieved during a research
stay of C.O. and D.P. at the Hausdorff Institute for
Mathematics in Bonn, which is thankfully acknowledged.

http://dx.doi.org/10.4171/OWR/2008/19

http://www.asc.tuwien.ac.at/~dirk/download/preprint/oberwolfach_abem.pdf