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].
| Identifer | oai:union.ndltd.org:NSYSU/oai:NSYSU:etd-0810110-200158 |
| Date | 10 August 2010 |
| Creators | Hsieh, Hsiu-Chen |
| Contributors | Zi-Cai Li, Tzon-Tzer Lu, Hung-Tsai Huang, Chien-Sen Huang, Ming-Gong Lee |
| Publisher | NSYSU |
| Source Sets | NSYSU Electronic Thesis and Dissertation Archive |
| Language | English |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | http://etd.lib.nsysu.edu.tw/ETD-db/ETD-search/view_etd?URN=etd-0810110-200158 |
| Rights | not_available, Copyright information available at source archive |
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