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M. Aurada, M. Feischl, M. Karkulik, D. Praetorius:
"Adaptive coupling of FEM and BEM: Simple error estimators and convergence (IABEM 2011)";
Talk: IABEM 2011 Conference, Brescia; 09-05-2011 - 09-08-2011; in: "Proceedings of IABEM 2011", (2011), 35 - 40.

A posteriori error estimators and adaptive mesh-refinement have themselves
proven to be important tools for scientific computing. For error control in finite element
methods (FEM), there is a broad variety of a posteriori error estimators available, and
convergence as well as quasi-optimality of adaptive FEM is well-studied in the literature,
cf. e.g. [Ainsworth et al. 2000] for error estimation and [Cascon et al. 2008] and the
references therein for convergence and quasi-optimality. This is, however, in sharp
contrast to the boundary element method (BEM) and the coupling of FEM and BEM,
cf. [Carstensen et al. 2001] for an overview on BEM error estimators and
[Ferraz-Leite et. al 2010] and [Aurada et al. 2010] for first preliminary convergence
results.

In our contribution, based on [Aurada et al. 2010], we present an easy-to-implement
(h − h/2)-type error estimator \mu for some FEM-BEM coupling which, to the best of
our knowledge, has not been proposed in the literature before. The considered
(h−h/2)-based approach is mathematically unifying in the sense that only stability of
the FEM-BEM coupling as well as certain inverse estimates and approximation estimates
for the energy norm are used. It is therefore applicable to symmetric as well as
non-symmetric FEM-BEM formulations without any modification.

The lower bound \mu <= C*error does always hold, whereas the upper bound error
error <= C*\mu depends on a saturation assumption. In numerical experiments, this
assumption, which is mathematically crucial, is empirically checked and verified.

The proposed mesh-refining algorithm provides the first adaptive coupling procedure
which is mathematically proven to converge. More precisely, we show that the adaptive
algorithm, based on Doerfler marking [Doerfler 1996] and newest vertex bisection,
drives the underlying error estimator to zero.

http://www.asc.tuwien.ac.at/~dirk/download/published/afkp_iabem2011.pdf