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  • Applications of wavelet bases to numerical solutions of elliptic equations
  • Zhao, Wei
  • en
  • wavelets, Riesz bases, elliptic equations, multilevel decompositions, splines
  • Jul 20, 2010 2:33 PM
  • Thesis
  • en
  • Adobe PDF
  • 1868299 bytes
  • 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.
  • Doctoral
  • Doctor of Philosophy
  • Department of Mathematical and Statistical Sciences
  • Fall 2010
  • Jia, Rong-Qing (Mathematical and Statistical Sciences)
  • Jia, Rong-Qing (Mathematical and Statistical Sciences)
    Wong, Yau Shu (Mathematical and Statistical Sciences)
    Han, Bin (Mathematical and Statistical Sciences)
    Derksen, Jos (Chemical and Materials Engineering)
    Xu, Yuesheng (Mathematics)


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