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博碩士論文 etd-0802105-042131 詳細資訊
Title page for etd-0802105-042131
論文名稱
Title
用非協調有限元求解特徵值問題
Non-conforming Finite Element Methods for Eigenvalue Problems
系所名稱
Department
畢業學年期
Year, semester
語文別
Language
學位類別
Degree
頁數
Number of pages
50
研究生
Author
指導教授
Advisor
召集委員
Convenor
口試委員
Advisory Committee
口試日期
Date of Exam
2005-05-26
繳交日期
Date of Submission
2005-08-02
關鍵字
Keywords
非協調元、雙線性元、特徵值問題、Wilson's 元、線性元、協調元、拓廣旋轉元、旋轉元
extension of rotated bilinear element, rotated bilinear element, Wilson's element, bilinear element, linear element, non- conforming, eigenvalue, conforming
統計
Statistics
本論文已被瀏覽 5708 次,被下載 2185
The thesis/dissertation has been browsed 5708 times, has been downloaded 2185 times.
中文摘要
本文探討 -Δu =λρu 的特徵值之有限元解的展開式,其中包括二個協調元:線性元 $P_1$ 和雙線性元 $Q_1$,以及三個非協調元:旋轉元 $Q_1^{rot}$ ,拓廣旋轉元 $EQ_1^{rot}$ 和 Wilson's 元。

根據此展開式,$P_1$ ,$Q_1$ 和 $Q_1^{rot}$ 給出特徵值的上界,而 $EQ_1^{rot}$ 和 Wilson's 元則給出特徵值的下界。對於最小特徵值而言,以 $Q_1^{rot}$ 元較精確。此外用外推法可以達到 $O(h^4)$ 的四階超收斂,其中 $h$ 是均勻方形的邊界長度。本文所作的數值實驗驗證了以上理論分析的結果。
Abstract
The thesis explores the new expansions of eigenvalues for -Δu =λρu in S with the Dirichlet boundary condition u=0 on $partial S$ by two conforming elements: the linear element $P_1$ and the bilinear element $Q_1$, and three non-conforming elements: the rotated bilinear element (denoted $Q_1^{rot}$), the extension of $Q_1^{rot}$ (denoted $EQ_1^{rot}$) and Wilson's element. The expansions indicate that $P_1$, $Q_1$ and $Q_1^{rot}$ provide the upper bounds of the eigenvalues, and $EQ_1^{rot}$ and Wilson's elements provide the lower bounds of the eigenvalues. Comparing the five finite elements, the $Q_1^{rot}$ element is more accurate. By the extrapolation, the superconvergence $O(h^4)$ can be obtained where $h$ is the boundary length of uniform squares. Numerical experiment are carried to verify the theoretical analysis made.
(參照電子檔p.4)
目次 Table of Contents
1 Introduction
2 Basis Functions and Algorithms
2.1 Linear element $P_1$ for Courant triangular mesh.
2.2 Bilinear element $Q_1$
2.3 Rotated $Q_1$ element, $Q_1^{rot}$
2.4 Extension of rotated $Q_1$ element, $EQ_1^{rot}$
2.5 Wilson's element
3 Error Expansion and Extrapolation Method
3.1 Error expansion method
3.2 Extrapolation method
4 Numerical Results
4.1 Function ρ = 1
4.2 Function ρ ≠ 1
5 Summary
參考文獻 References
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[2] J. Brandts and M. Krizek, History and future of superconvergence in three- dimen-sional FEMs, GAKUTO Inter. Series, Math. Sci., Appl., vol. 15, Tokyo, pp. 24-35, 2001.
[3] Z.C. Li and N. Yang, New error estimates of bi-cubic Hermite finite element methods for biharmonic equations, J., Comp. and Applied. Math., vol. 142, pp. 251-285, 2002.
[4] Q. Lin, High Performance FEMs, International Symposium on Computational and Applied PDEs, July 1-7, 2001, China.
[5] Q. Lin, H.T. Huang and Z.C. Li, New expansions of eigenvalues for -Δu =λρu by non-conforming elements, Technical report, Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 2005.
[6] Q. Lin and J. Lin, High Performance FEMs, to appear in China Sci. Tech. Press, 2003.
[7] Q. Lin and N. Yang, Construction and Analysis of E.ective FEMs, Hebei Univ. Press, China, 1996.
[8] Q. Lin and T. Lu, Asymptotic expansions for finite element eigenvalues and finite element solution, Proceedings 6 Int. Conf. on Comp. Meth. Appl. Sci. Eng. Versailles, 1983.
[9] D.S. Wu, Convergence and superconvergence of Hermite bicubic element for eigen-value problem of biharmonic equation, J. Comp. Math., vol. 19, pp. 139-142, 2001.
[10] J. Xu and A. Zhou, A two-grid discretization scheme for eigenvalue problems,Math. Comp., vol. 70, pp. 17-25, 1999.
[11] Y.D. Yang, Computable error bounds for an eigenvalue problem in the finite element method, Chinese Numer. Math. and Appl., vol. 17, No. 1, pp. 68-77, 1995.
[12] Y.D. Yang, A posteriori error estimates in Adini finite element for eigenvalue prob-lems, J. Comp. Math., vol. 18, pp. 413-418, 2000.
[13] R. Rannacher and S. Turek, Simple nonconforming quadrilateral Stokes element,
Numer. Methods PDEs, Vol. 8, pp. 97-111, 1992.
[14] Q. Lin, L. Tobiska and A. Zhou, On the superconvergence of nonconforming low
order finite elements applied to the Poisson equation, Preprint, 2001.
[15] T. Lu, T.M. Shih and C.B. Liem, The Splitting Extrapolation Method and
Combination Techniques, Science Press, China, 1998.
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