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Applications of wavelet bases to numerical solutions of elliptic equations Open Access

Descriptions

Other title
Subject/Keyword
wavelets, Riesz bases, elliptic equations, multilevel decompositions, splines
Type of item
Thesis
Degree grantor
University of Alberta
Author or creator
Zhao, Wei
Supervisor and department
Jia, Rong-Qing (Mathematical and Statistical Sciences)
Examining committee member and department
Xu, Yuesheng (Mathematics)
Han, Bin (Mathematical and Statistical Sciences)
Derksen, Jos (Chemical and Materials Engineering)
Wong, Yau Shu (Mathematical and Statistical Sciences)
Jia, Rong-Qing (Mathematical and Statistical Sciences)
Department
Department of Mathematical and Statistical Sciences
Specialization

Date accepted
2010-07-20T14:33:09Z
Graduation date
2010-11
Degree
Doctor of Philosophy
Degree level
Doctoral
Abstract
In this thesis, we investigate Riesz bases of wavelets and their applications to numerical solutions of elliptic equations. Compared with the finite difference and finite element methods, the wavelet method for solving elliptic equations is relatively young but powerful. In the wavelet Galerkin method, the efficiency of the numerical schemes is directly determined by the properties of the wavelet bases. Hence, the construction of Riesz bases of wavelets is crucial. We propose different ways to construct wavelet bases whose stability in Sobolev spaces is then established. An advantage of our approaches is their far superior simplicity over many other known constructions. As a result, the corresponding numerical schemes are easily implemented and efficient. We apply these wavelet bases to solve some important elliptic equations in physics and show their effectiveness numerically. Multilevel algorithm based on preconditioned conjugate gradient algorithm is also developed to significantly improve the numerical performance. Numerical results and comparison with other existing methods are presented to demonstrate the advantages of the wavelet Galerkin method we propose.
Language
English
Rights
License granted by Wei Zhao (wazhao@math.ualberta.ca) on 2010-07-17T01:14:37Z (GMT): Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of the above terms. The author reserves all other publication and other rights in association with the copyright in the thesis, and except as herein provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.
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