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M. Page:
"Schätzerreduktion und Konvergenz adaptiver FEM für Hindernisprobleme";
Supervisor: D. Praetorius; Institute for Analysis and Scientific Computing, 2010; final examination: 2010-05-20.

This thesis deals with the construction of adaptive algorithms that
solve elliptic obstacle problems by means of the nite element method.
Throughout all chapters, the main idea to do so is the generalization
of the so-called estimator reduction to such (nonlinear) problems. In
this way, it is possible to widen and improve known results. One of
the ingredients frequently used in the convergence analysis of
adaptive nite element methods is the discrete local efficiency
of the underlying error estimator. This feature, however, strongly
relies on the interior-node property which basically states that at
least one new node has to be created within the interior of each
refi ned element. All convergence results exploiting the discrete
local efficiency thus depend on the local re nement strategy and are
therefore non-universal. By means of the method of estimator
reduction from [CKNS, AFP] it is possible though, to circumvent the
need for this property and thus achieve convergence results that are
independent of the local mesh re nement strategy. In this thesis, this
ansatz, which is known from the linear case, is pursued in order to
generalize the results from [BCH1] and [BCH2]. Throughout the whole
work, the two dimensional Poisson-equation serves as model problem.
Generalizations to the three-dimensional case are, however, easily
possible. In full detail, the work is organized as follows:

Chapter 1 gives an introduction to the problem and comments on some
physical interpretation of the obstacle problems in question.

In Chapter 2, the fi nite element method (FEM) is explained in full
detail. The better part of the used notation as well as some basic
ideas used in the upcoming proofs are also illustrated in this linear
setting. The main result within this chapter is Theorem 2.2.6 which
states the principle of estimator reduction for the linear case.

A generalization of the estimator reduction to obstacle problems in
which the obstacle is globally affine, is given in Chapter 3. As
mentioned above, an adaptive FEM-algorithm and a corresponding a
posteriori error estimator is given. The convergence towards the
exact solution (Theorem 3.3.9) is shown via estimator reduction
(Theorem 3.3.3) and is therefore independent of the local mesh
refinement strategy. By utilizing this approach, some postulations
on the decay of oscillation data from [BCH1] additionally become
unneccessary. In order to fully appreciate the advantages of this
ansatz, the original ideas from [BCH1] are elaborated as well.

In Chapter 4 the analysis is widened to include obstacles from H^1_0.
As before, the the results from [BCH2], which make use of the
discrete local efficiency, are elaborated. In the following, the
results are again generalized by estimator reduction and a convergent
adaptive algorithm with a corresponding error estimator is given.
During the convergence analysis (Theorem 4.4.6 { 4.4.8), however, the
need for some additional assumptions on the change of active and
inactive regions, arises. Certain limits of the principle of
estimator reduction, that could not be foreseen within the linear
setting,can therefore be glimpsed. In addition, Chapter 3, as well as
Chapter 4 are completed by some numerical illustrations.

Chapter 5 gives a conclusion about achieved results and some outlook
on possible future work.

Finally, a complete Matlab implementation of the numerical examples
is given in the appendix (Anhang). A list of notations as well as
some mathematical background information can be found here, too.

http://www.asc.tuwien.ac.at/~dirk/download/thesis/msc/page2010.pdf