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M. Karkulik:
"Zur Konvergenz und Quasioptimalität adaptiver Randelementmethoden";
Supervisor, Reviewer: D. Praetorius, J. Melenk, S. Sauter; Institute for Analysis and Scientific Computing, 2012; oral examination: 2012-10-29.

The purpose of this work is to develop a convergence theory for
adaptive Galerkin methods for boundary integral equations. For
adaptive Finite Element Methods, such a theory was developed in the
last 15 years. However, the ideas cannot be used directly in boundary
element methods, as the underlying operators and norms are non-local
in their nature.

Chapter 3 deals with meshes and spaces of piecewise polynomials,
which are used for the Galerkin discretization of the underlying
problems. The existing results on the optimality of the mesh-closure
in case of local mesh-refinement are extended. Hence the conditions,
under which adaptive Finite Element Methods are guaranteed to
converge, are relaxed. Furthermore, (local) approximation properties
in Sobolev space of non-integer order are shown for various
quasi-interpolation operators. In the following, the boundary
integral equations arising from the Laplace equation are used as model
problems. The underlying boundary integral operators are introduced
in chapter 3, and novel inverse inequalities involving these
non-local operators are shown. To that end, results of elliptic
regularity theory are exploited.

Chapter 5 is concerned with the convergence and quasi-optimality of
adaptive Galerkin methods for equations involving the simple layer
potential. We use two different approaches for a-posteriori error
estimation. First, we consider h−h/2-based error estimators, where
solutions to uniformly refined meshes are used to estimate the error.
Second, we deal with weighted residual error estimators, which use
the equations´ residual to estimate the error. In case of the
Dirichlet problem, we additionally approximate the given boundary
data by discrete functions. To that end, we use certain
quasi-interpolation operators, which were analyzed in chapter 3. We
prove convergence of adaptive algorithms steered by either
h−h/2-based error estimators or weighted residual error estimators.
To that end, we use the concept of estimator reduction: to show
convergence of the estimator, it suffices to show that it is a
contraction up to a zero sequence. In case of the weighted residual
estimator, we will employ the inverse inequalities for boundary
integral operators. Furthermore, we prove the quasi-optimality of
adaptive algorithms steered by weighted residual error estimators. By
that, we mean the following: assume that there is an arbitrary
sequence of meshes, starting from a fixed coarse mesh, such that the
error estimator converges with a certain rate. Under this assumption,
we show that the proposed adaptive algorithm exhibits the same rate
for the underlying error estimator. To prove these kind of
statements, we use tools from the convergence theory for adaptive
Finite Element Methods, which we transfer to the case of non-local
operators and norms.

In chapter 6, we show the convergence and quasi-optimality of
adaptive algorithms for equations involving the hypersingular
integral operator. We use the same approaches as in chapter 5.

Chapter 7 contains several numerical experiments for equations with
the weakly singular integral operator. We consider Symm´s integral
equation and the Dirichlet problem for the Laplace equation. The
experiments support the developed analytical results and show the
efficiency of the adaptive algorithms.

http://www.asc.tuwien.ac.at/~dirk/download/thesis/phd/karkulik2012.pdf