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M. Feischl, G. Gantner, D. Praetorius:
"A posteriori error estimation for adaptive IGA boundary element methods";
Talk: 11th World Congress on Computational Mechanics (WCCM XI), Barcelona; 07-20-2014 - 07-25-2014; in: "11th World Congress on Computational Mechanics (WCCM XI)", (2014), 2421 - 2432.

A posteriori error estimation and adaptive mesh-refinement are well-established and
important tools for standard boundary element methods (BEM) for polygonal boundaries and
piecewise polynomial ansatz functions. In contrast, the mathematically reliable a posteriori
error analysis for isogeometric BEM (IGABEM) has not been considered, yet. In our talk, we aim
to shed some light on this gap and to transfer known results on reliable a posteriori error
estimators [Faermann 2000, 2002] from standard BEM to IGABEM.

As model example serves the weakly-singular integral equation for the 2D Laplacian. For our
IGABEM, we employ non-uniform rational B-splines (NURBS). We prove that the (numerically
computable) Faermann error estimator provides lower and upper bounds for the (in general,
non-computable and unknown) error in the H^{-1/2} energy norm. We prove that the
constants involved remain bounded even if the mesh is locally refined by node-insertion. In
particular, the error can thus be used to monitor the error decay if the mesh is refined.
Moreover, its local contributions can be used for adaptive IGABEM computations to steer an
adaptive algorithms of the form

SOLVE -> ESTIMATE -> MARK -> REFINE

which automatically detect singularities of the solution and adapt the mesh accordingly. If
compared to uniform mesh refinement, this dramatically reduces the storage requirements
as well as the computing time needed to achieve a certain prescribed accuracy.

http://www.wccm-eccm-ecfd2014.org/frontal/Ebook.asp