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.

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.

http://www.asc.tuwien.ac.at/preprint/2011/asc21x2011.pdf

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