[Back]

M. Feischl, M. Karkulik, J. Melenk, D. Praetorius:
"Convergence and quasi-optimality of adaptive boundary element methods";
Talk: Efficient mesh adaptation methods for evolution problems: theory and applications, Wolfgang-Pauli-Institut Wien (invited); 12-14-2011 - 12-17-2011.

A posteriori error estimation and adaptive mesh-refinement have
themselves proven to be effective tools in scientific computing. 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 mathematically
explained for adaptive finite element methods (AFEM) and elliptic
model problems. However, even plain convergence of AFEM has been a
major topic in research.

For boundary integral equations, the fractional Sobolev spaces and
the non-local boundary integral operators involved lead to further
technical difficulties. Consequently, for adaptive boundary element
methods (ABEM) much less is known and even convergence of ABEM has
essentially been open.

In our talk, we consider an adaptive algorithm in the context of the
boundary elment method (ABEM), where the mesh-refinement is driven by
the weigthed-residual error estimator. We discuss the estimator
reduction concept to prove convergence of this ABEM. Moreover, we
provide the mathematical frame 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
class involved.