Title page for etd-0526104-190036


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URN etd-0526104-190036
Author Hsin-Yun Hu
Author's Email Address No Public.
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Department Applied Mathematics
Year 2003
Semester 2
Degree Ph.D.
Type of Document
Language English
Title The Trefftz and Collocation Methods for Elliptic Equations
Date of Defense 2004-04-28
Page Count 189
Keyword
  • collocation method
  • elliptic equations
  • Trefftz method
  • singularities
  • combined method
  • Abstract The dissertation consists of two parts.The first part is mainly to provide the algorithms and error estimates of the collocation Trefftz methods (CTMs) for seeking the solutions of partial differential equations. We consider several popular models of PDEs with singularities, including Poisson equations and the biharmonic equations. The second part is to present the collocation methods (CMs) and to give a unified framework of combinations of CMs with other numerical methods such as finite element method, etc. An interesting fact has been justified: The integration quadrature formulas only affect on the uniformly $V_h$-elliptic inequality, not on the solution accuracy. In CTMs and CMs, the Gaussian quadrature points will be chosen as the collocation points. Of course, the Newton-Cotes quadrature points can be applied as well. We need a suitable dense points to guarantee the uniformly $V_h$-elliptic inequality. In addition, the solution domain of problems may not be confined in polygons. We may also divide the domain into several small subdomains. For the smooth solutions of problems, the different degree polynomials can be chosen to approximate the solutions properly. However, different kinds of admissible functions may also be used in the methods given in this dissertation. Besides, a new unified framework of combinations of CMs with other methods will be analyzed. In this dissertation, the new analysis is more flexible towards the practical problems and is easy to fit into rather arbitrary domains. Thus is a great distinctive feature from that in the existing literatures of CTMs and CMs. Finally, a few numerical experiments for smooth and singularity problems are provided to display effectiveness of the methods proposed, and to support the analysis made.
    Advisory Committee
  • Tsu-Fen Chen - chair
  • Jinn-Liang Liu - co-chair
  • Wen-Wei Lin - co-chair
  • Cheng-Sheng Chien - co-chair
  • Zi-Cai Li - advisor
  • Files
  • etd-0526104-190036.pdf
  • indicate access worldwide
    Date of Submission 2004-05-26

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