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G. Mitscha-Eibl:
"Adaptive BEM und FEM-BEM-Kopplung für die Lamé-Gleichung";
Supervisor: T. Führer, D. Praetorius; Institute for Analysis and Scientific Computing, 2014; final examination: 2014-10-16.

We are concerned with the analysis and numerical solution of some boundary value
problems for the Lamé equation from linear elasticity. These partial di erential equations
are reformulated as boundary integral equations and are then treated numerically by use
of BEM (boundary element method) and the coupling of BEM and FEM ( finite element
method).

A special focus of our work lies on adaptivity: We use error indicators to get local information
on where to refi ne the underlying boundary mesh and thereby to increase the accuracy of the
discrete approximate solution. We introduce and analyze various error indicators for adaptive
BEM and FEM-BEM-coupling. An important tool in our analysis is the theory of interpolation
spaces, see [BL76]. We contribute to this theory by considering interpolation of semi-norms
instead of norms. We prove equivalence of the interpolation semi-norm and the usual
Slobodeckij seminorm on fractional-order Sobolev spaces Hs, and study how the
involved equivalence constants behave under Lipschitz transformations of the domain X,
thereby generalizing results from the recent paper [Heu14]. Next, we use this to prove
uniform equivalence of the two kinds of semi-norms on node patches of a triangulated
surface. As an application of our findings in interpolation space theory, we give a new proof
of the localisation of the Hs-norm from [CMS01], a fundamental result to establish, for
instance, reliability of a weighted-residual error estimator for BEM.

Numerical experiments in 2D for BEM as well as FEM-BEM-coupling are presented to
con firm our theoretical findings and illustrate their applicability. A detailed discussion of
implementational issues such as numerically stable computation of the discrete boundary
integral operators is included as well.

http://www.asc.tuwien.ac.at/~dirk/download/thesis/msc/mitschaeibl2014.pdf