M. Aurada, M. Feischl, M. Karkulik, D. Praetorius:

"Adaptive coupling of FEM and BEM: Simple error estimators and convergence (IABEM 2011)";

in: "ASC Report 20/2011", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2011, ISBN: 978-3-902627-04-9.

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/preprint/2011/asc20x2011.pdf

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