Collocation Fourier methods for Elliptic andEigenvalue Problems
碩士 === 國立中山大學 === 應用數學系研究所 === 98 === 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 t...
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ndltd-TW-098NSYS55070862015-10-13T18:39:46Z http://ndltd.ncl.edu.tw/handle/32303062094290545645 Collocation Fourier methods for Elliptic andEigenvalue Problems 利用傅立葉配置法求解橢圓及特徵值問題 Hsiu-Chen Hsieh 謝秀真 碩士 國立中山大學 應用數學系研究所 98 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]. Zi-Cai Li Hung-Tsai Huang 李子才 黃宏財 2010 學位論文 ; thesis 97 en_US |
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碩士 === 國立中山大學 === 應用數學系研究所 === 98 === 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].
|
author2 |
Zi-Cai Li |
author_facet |
Zi-Cai Li Hsiu-Chen Hsieh 謝秀真 |
author |
Hsiu-Chen Hsieh 謝秀真 |
spellingShingle |
Hsiu-Chen Hsieh 謝秀真 Collocation Fourier methods for Elliptic andEigenvalue Problems |
author_sort |
Hsiu-Chen Hsieh |
title |
Collocation Fourier methods for Elliptic andEigenvalue Problems |
title_short |
Collocation Fourier methods for Elliptic andEigenvalue Problems |
title_full |
Collocation Fourier methods for Elliptic andEigenvalue Problems |
title_fullStr |
Collocation Fourier methods for Elliptic andEigenvalue Problems |
title_full_unstemmed |
Collocation Fourier methods for Elliptic andEigenvalue Problems |
title_sort |
collocation fourier methods for elliptic andeigenvalue problems |
publishDate |
2010 |
url |
http://ndltd.ncl.edu.tw/handle/32303062094290545645 |
work_keys_str_mv |
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