[Back]

M. Feischl, M. Karkulik, J. Melenk, D. Praetorius:
"Quasi-optimal convergence rate for an adaptive boundary element method";
Talk: BEM on the Saar 2012, Universität des Saarlandes; 05-12-2012 - 05-16-2012.

A posteriori error estimation and adaptive mesh-refinement are effective tools in scientific com-
puting. In practice, these algorithms lead to quasi-optimal convergence behaviour with respect to the
number of degrees of freedom. In the last decade, these empirical observations have been put on a firm
mathematical foundation for adaptive finite element methods and elliptic model problems in 2D.
For boundary integral equations, the fractional Sobolev spaces and the non-local boundary integral
operators involved preclude a direct transfer of the FEM techniques. Thus, for adaptive boundary ele-
ment methods (ABEM) much less is known and even convergence of ABEM has essentially been open.
Our prior works on convergence of ABEM considered h-h/2-based error estimators. Reliability of
these type of estimators is, however, equivalent to the so-called saturation assumption.
In this talk, which follows the recent work [2], we consider an adaptive algorithm in the context of the
boundary element method (ABEM), where the mesh-refinement is driven by the weighted residual error
estimator of [1]. This estimator enjoys the property to be reliable, without appealing to the saturation
assumption. We discuss the estimator reduction concept to prove convergence of this ABEM. Moreover,
we provide the mathematical framework to prove quasi-optimal convergence rates. Emphasis is laid on
the fact that the efficiency of the error estimator (lower bound) is not needed to prove quasi-optimality
of the adaptive algorithm, but only to characterize the approximation classes.