M. Page:

"Schätzerreduktion und Konvergenz adaptiver FEM für Hindernisprobleme";

Supervisor: D. Praetorius; Institute for Analysis and Scientific Computing, 2010; final examination: 05-20-2010.

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

Created from the Publication Database of the Vienna University of Technology.