Contributions to Books:

M. Feischl, M. Karkulik, J. Melenk, D. Praetorius:
"Residual a-posteriori error estimates in BEM: Convergence of h-adaptive algorithms";
in: "ASC Report 21/2011", issued by: Institut for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2011, ISBN: 978-3-902627-04-9.

English abstract:
Galerkin methods for FEM and BEM based on uniform mesh refinement
have a guaranteed rate of convergence. Unfortunately, this rate may be suboptimal due
to singularities present in the exact solution. In numerical experiments, the optimal rate
of convergence is regained when algorithms based on a-posteriori error estimation and
adaptive mesh-refinement are used. This observation was proved mathematically for the
FEM in the last few years, cf. [5]. In constrast, the mathematical understanding of
adaptive strategies is wide open in BEM. One reason for this is the non-locality of the
boundary integral operators involved and the appearance of fractional-order or negative
Sobolev norms.
In our prior works on adaptive BEM [1], we considered h−h/2 error estimators. Reliability
of such estimators is, however, equivalent to the so-called saturation assumption.
Although this is widely believed to hold in practice, it still remains mathematically open.
For this reason, these convergence results are not fully satisfactory.
In our talk, we consider weighted-residual error estimators for some weakly-singular
integral equations in 2D or 3D. These estimators are reliable, irrespective of the saturation
assumption. We prove a certain (local) inverse-type estimate which allows us to
conclude that the discrete solutions generated by the usual h-adaptive algorithm converge
towards the exact solution of the integral equation. In a second step we prove
quasi-optimality in a certain approximation class. From this, we infer that the rate of
convergence of adaptive mesh-refinement is at least as good as for uniform approaches.

Electronic version of the publication:

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