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

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