Title page for etd-0819111-172202


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URN etd-0819111-172202
Author Sin-Rong Wu
Author's Email Address No Public.
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Department Applied Mathematics
Year 2010
Semester 2
Degree Master
Type of Document
Language English
Title The Collocation Trefftz Method for Laplace's Equation on Annular Shaped Domains, Circular and Elliptic Boundaries
Date of Defense 2011-06-09
Page Count 125
Keyword
  • annular shaped domains
  • circular domains
  • null field method
  • Collocation Trefftz method
  • interior field method
  • fundamental solutions
  • error analysis
  • Dirichelet condition
  • Abstract The collocation Trefftz method (CTM) proposed in [36] is employed to annular shaped domains, and new error analysis is made to yield the optimal convergence rates. This popular method is then applied to the special case: the Dirichlet problems on circular domains with circular holes, and comparisons are made with the null field method (NFM) proposed , and new interior field method (IFM) proposed in [35], to find out that both
    errors and condition numbers are smaller.
    Recently, for circular domains with circular holes, the null fields method (NFM) is proposed by Chen and his groups. In NFM, the fundamental solutions (FS) with the source nodes Q outside of the solution domains are used in the Green formulas, and the FS are replaced by their series expansions. The Fourier expansions of the known or the unknown Dirichlet and Neumann boundary conditions on the circular boundaries are chosen, so that the explicit discrete equations can be easily obtained by means of orthogonality of Fourier functions. The NFM has been applied to elliptic equations and eigenvalue problems in circular domains with multiple holes, reported in many papers; here we cite those for Laplace’s equation only (see [18, 19, 20]). For the boundary integral equation (BIE) of the first kind, the trigonometric functions are used in Arnold [4, 5], and error analysis is made for infinite smooth solutions, to derive the exponential convergence rates. In Cheng’s Dissertation [21, 22], for BIE of the first kind, the source nodes are
    located outside of the solution domain, the linear combination of fundamental solutions are used, and error analysis is made only for circular domains.
    This fact implies that not only can the CTM be applied to arbitrary domains, but also a better numerical performance is provided. Since the algorithms of the CTM is simple and its programming is easy, the CTM is strongly recommended to replace the NFM for circular domains with circular holes in engineering problems.
    Advisory Committee
  • Chien-Sen Huang - chair
  • Tsung-Lin Lee - co-chair
  • Ming-Gong Lee - co-chair
  • Jeng-Tzong Chen - co-chair
  • Zi-Cai Li - advisor
  • Files
  • etd-0819111-172202.pdf
  • Indicate in-campus at 1 year and off-campus access at 1 year.
    Date of Submission 2011-08-19

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