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M. Feischl, M. Karkulik, J. Melenk, G. Of, D. Praetorius:
"A survey on adaptive boundary element methods";
Talk: Fast BEM and BETI, Ostrava (Tschechien) (invited); 06-18-2012 - 06-19-2012.

Adaptivity has become a fundamental instrument for the efficient numerical solution of partial differential equations.
In traditional FEM/BEM analysis, meshes are often designed based on a priori information and experience.
In contrast to that, adaptive procedures try to automatically refine or coarsen a mesh or adjust the underlying basis functions to achieve a good solution:
Starting with a given initial mesh $\TT_0$ and based on certain refinement indicators, usual adaptive algorithms of the type

solve -> estimate -> mark -> refine

mark elements $\MM_\ell\subseteq\TT_\ell$ for refinement, use a refinement rule to generate a mesh $\TT_{\ell+1}:={\tt refine}(\TT_\ell,\MM_\ell)$,
where at least the marked elements $T\in\MM_\ell$ are refined, and iterate.

In the presence of singularities of the unknown exact solution or given data, uniform meshes are suboptimal.
However, adaptive methods regain the optimal rate of convergence in numerical experiments.

For finite element methods, adaptive algorithms and their different components have been studied since the early 1980s, starting with the pioneering works
of Babuska on a-posteriori error estimates. The last decade saw a major breakthrough in the mathematical understanding of quasi-optimality of adaptive FEM with the works [Binev/Dahmen/deVore '04] and [Cascon/Kreuzer/Nochetto/Siebert '08].
In BEM, the situation is less developed. The underlying integral operators and the natural norms are non-local, which induces additional difficulties in the design
of a-posteriori error estimates.

In this talk, we give a short overview on the state of the art in adaptive BEM.
We present certain a-posteriori error estimate, and discuss the design of the different components of the adaptive algorithm.
We will comment on the quasi-optimality and give some
numerical examples to illustrate the performance of adaptive BEM.