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M. Feischl:
"Optimality of adaptive 2D boundary element method";
Supervisor: M. Karkulik, D. Praetorius; Institute for Analysis and Scientific Computing, 2012; final examination: 03-22-2012.

Recently, there was a major breakthrough in the mathematical
understanding of convergence and quasi-optimality of h-adaptive FEM
for second-order elliptic PDEs. However, many of the ingredients
which appear in the proofs were mathematically open for adaptive BEM.

In Chapter 2 we consider a general adaptive algorithm (e.g. BEM or
FEM) and work out these ingredients to develop a fully abstract
framework for proving convergence and quasi-optimality of general
adaptive algorithms of the type:

solve −> estimate −> mark −> refine

Moreover, we formulate a set of sufficient assumptions, which are
used to prove the three main results of this work:

o convergence of the adaptive algorithm,
o optimal convergence rate of the estimator,
o characterization of the approximation class and therefore optimal
convergence rate of the error.

In contrast to prior works on optimality of adaptive algorithms, we
show that efficiency of the error estimator is only needed to
characterize the approximation class, whereas convergence and
optimality of the adaptive algorithm mainly depend on discrete local
reliability of the estimator. Chapter 4 applies the abstract analysis
to a concrete model problem, i.e. for a polygonal Lipschitz domain
Omega im R2, we analyze Symm´s integral equation

V phi = (K + 1/2) g

on the boundary of Gamma for some given boundary data. We use the
weighted-residual error estimator from [Carstensen, Stephan 1996] to
steer the mesh refinement. We prove discrete local reliability and a
new inverse estimate, which allows us to prove convergence of the
adaptive algorithm. To get optimality of the adaptive algorithm, we
present two 1D mesh refining strategies in Section 4.4 which
guarantee uniform shape regularity of the constructed meshes and
satisfy several other properties which are needed to apply the
abstract analysis of Chapter 2. Whereas, e.g. newest vertex bisection
fulfills all required properties for meshes in 2D and 3D, the proof
in 1D is inherently different, because we cannot rely on the angle
condition to prevent the collapse of an element, but we need to bound
the ratio of the diameters of two neighboring elements. Under some
additional regularity assumptions on the boundary data g, we are able
to prove efficiency of the weighted-residual error estimator on
locally refined meshes, which was priorly only known on quasi uniform
meshes under even slightly stronger regularity assumptions, see
[Carstensen 1996]. Chapter 5 incorporates the approximation
of the Dirichlet data g by a discrete function G` in each step of the
adaptive algorithm. Using the concept of modified D¨orfler marking
which goes back to [Stevenson 2007], we prove convergence and
quasi-optimality of the corresponding adaptive algorithm. Finally,
Chapter 6 provides some numerical experiments, which underline the
results of this work and give a comparison to naive uniform
mesh-refinement. We conclude the work with some remarks on the
saturation assumption and give a slightly weaker result in the
Appendix.

http://www.asc.tuwien.ac.at/~dirk/download/thesis/msc/feischl2012.pdf