Title page for etd-0810110-200158


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URN etd-0810110-200158
Author Hsiu-Chen Hsieh
Author's Email Address No Public.
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Department Applied Mathematics
Year 2009
Semester 2
Degree Master
Type of Document
Language English
Title Collocation Fourier methods for Elliptic and
Eigenvalue Problems
Date of Defense 2010-06-03
Page Count 97
Keyword
  • eigenvalue problems
  • Bose-Einstein condensation
  • periodic potential
  • stability analysis
  • energy level
  • spectral method
  • elliptic problems
  • Error analysis
  • Abstract In spectral methods for numerical PDEs, when the solutions are periodical, the Fourier
    functions may be used. However, when the solutions are non-periodical, the Legendre and
    Chebyshev polynomials are recommended, reported in many papers and books. There
    seems to exist few reports for the study of non-periodical solutions by spectral Fourier
    methods under the Dirichlet conditions and other boundary conditions. In this paper, we
    will explore the spectral Fourier methods(SFM) and collocation Fourier methods(CFM)
    for elliptic and eigenvalue problems. The CFM is simple and easy for computation, thus
    for saving a great deal of the CPU time. The collocation Fourier methods (CFM) can
    be regarded as the spectral Fourier methods (SFM) partly with the trapezoidal rule.
    Furthermore, the error bounds are derived for both the CFM and the SFM. When there
    exist no errors for the trapezoidal rule, the accuracy of the solutions from the CFM is as
    accurate as the spectral method using Legendre and Chebyshev polynomials. However,
    once there exists the truncation errors of the trapezoidal rule, the errors of the elliptic
    solutions and the leading eigenvalues the CFM are reduced to O(h^2), where h is the
    mesh length of uniform collocation grids, which are just equivalent to those by the linear
    elements and the finite difference method (FDM). The O(h^2) and even the superconvergence
    O(h4) are found numerically. The traditional condition number of the CFM
    is O(N^2), which is smaller than O(N^3) and O(N^4) of the collocation spectral methods
    using the Legendre and Chebyshev polynomials. Also the effective condition number is
    only O(1). Numerical experiments are reported for 1D elliptic and eigenvalue problems,
    to support the analysis made. The simplicity of algorithms and the promising numerical
    computation with O(h^4) may grant the CFM to be competent in application in numerical
    physics, chemistry, engineering, etc., see [7].
    Advisory Committee
  • Tzon-Tzer Lu - chair
  • Ming-Gong Lee - co-chair
  • Chien-Sen Huang - co-chair
  • Zi-Cai Li - advisor
  • Hung-Tsai Huang - advisor
  • Files
  • etd-0810110-200158.pdf
  • indicate not accessible
    Date of Submission 2010-08-10

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