Talks and Poster Presentations (without Proceedings-Entry):

M. Feischl, T. Führer, M. Karkulik, J. Melenk, D. Praetorius:
"Quasi-Optimal Convergence Rates for Some Adaptive Boundary Element Method in 2D and 3D";
Talk: 7th Zürich Summerschool on A Posteriori Error Control and Adaptivity, Zürich; 08-20-2012 - 08-24-2012.

English abstract:
We prove convergence and quasi-optimality of some lowest-order adaptive
boundary element method for a weakly-singular integral equation in 2D and 3D.
The adaptive mesh-refinement is driven by the weighted-residual
error estimator.

We show that the error estimator fulfils an estimator reduction property in each step of the
adaptive algorithm. Thereof, we may conclude linear convergence of the estimator. Moreover, reliability states that the estimator
provides a upper bound of the Galerkin error. Therefore, we even obtain linear convergence of adaptive BEM.
By proving discrete reliability of the estimator, we are able to follow the steps
of adaptive FEM and obtain optimal convergence rates for the estimator. In the 2D case, we even prove, that under some regularity assumptions on the given data, this estimator is not only reliable, but also efficient. Finally, we use efficiency of the estimator to characterize the approximation class in terms of the Galerkin error only. In particular, this yields that in 2D, no adaptive strategy can do better, and the weighted-residual error estimator is thus an optimal choice to steer the adaptive mesh-refinement.

Created from the Publication Database of the Vienna University of Technology.