Adaptive Wavelet and Frame Schemes for Elliptic and Parabolic Equations
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Philipps-Universität Marburg
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Abstract
This thesis is concerned with the application of wavelet methods to the adaptive numerical solution of elliptic and parabolic operator equations over a polygonal domain.
Driven by the insight that the construction of wavelet bases on more general domains is complicated and may pose stability problems, we analyze the option to replace the concept of wavelet bases by the more flexible concept of wavelet frames. Frames are redundant systems that still allow for stable decomposition and reconstruction of a given function. In the first part of this thesis, is shown how to construct so-called Gelfand frames on polygonal domains by a simple overlapping domain decomposition approach. Gelfand frames are able to characterize function spaces in a similar way as in the case of wavelet bases.
The second part is concerned with the application of Gelfand frames to the adaptive numerical treatment of linear elliptic problems. We propose inexact versions of well-known iterative schemes for the frame coordinate representation of the given operator equation. Both convergence and optimality of the considered methods can be proved and illustrated by numerical examples.
In the third part, we consider adaptive wavelet methods for the numerical treatment of linear parabolic equations. Due to the initial value problem structure of the latter, we consider a semidiscretization in time with linearly implicit methods first. The arising sequence of elliptic operator equations is then solved adaptively with wavelet methods. It is shown how to exploit the key properties of wavelet bases to a considerable extent, e.g., in preconditioning strategies and for the convergence and complexity analysis of the overall algorithm. We finish with numerical experiments in one and two spatial dimensions.
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Dates
Created: 2007Issued: 2007-06-08Updated: 2011-08-10
Faculty
Fachbereich Mathematik und Informatik
Publisher
Philipps-Universität Marburg
Language
eng
Data types
DoctoralThesis
Keywords
Adaptive methodsWaveletsFramesFinite elements, Rayleigh-Ritz and Galerkin methods, finite methodsFramesParabolic problemElliptic problemSeries of general orthogonal functions, generalized Fourier expansions, nonorthogonal expansionsStability and convergence of numerical methodsComplexity and performance of numerical algorithmsWavelets
DFG-subjects
Adaptives VerfahrenParabolisches RandwertproblemNumerische MathematikWavelet-AnalyseElliptisches Randwertproblem
DDC-Numbers
510
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Raasch, Thorsten: Adaptive Wavelet and Frame Schemes for Elliptic and Parabolic Equations. : Philipps-Universität Marburg 2007-06-08. DOI: https://doi.org/10.17192/z2007.0343.
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This item has been published with the following license: In Copyright