M. Feischl:

"Optimality of adaptive 2D boundary element method";

Supervisor: M. Karkulik, D. Praetorius; Institute for Analysis and Scientific Computing, 2012; final examination: 2012-03-22.

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

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