M. Feischl,M. Karkulik, J. Melenk, D. Praetorius:

"Residual a-posteriori error estimates in BEM: convergence of h-adaptive algorithms";

Talk: IABEM 2011 Conference, Brescia; 09-05-2011 - 09-08-2011; in: "Proceedings of IABEM 2011", (2011), 135 - 140.

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. [Cascon, Kreuzer, Nochetto, Siebert, 2008]

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 [Aurada, Ferraz-Leite, Goldenits, Karkulik, Mayr, Praetorius, 2001], 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 estimator 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://publik.tuwien.ac.at/files/PubDat_198315.pdf

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