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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 |
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系所名稱 Department |
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畢業學年期 Year, semester |
語文別 Language |
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學位類別 Degree |
頁數 Number of pages |
50 |
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研究生 Author |
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指導教授 Advisor |
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召集委員 Convenor |
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口試委員 Advisory Committee |
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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 |
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統計 Statistics |
本論文已被瀏覽 5708 次,被下載 2185 次 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 |
[1] I. Babuska and J. E. Osborn, Eigenvalue problems in Finite Element Methods,Part I , Eds. by P.G. Ciarlet and J. L. Lions, Elsevier, pp. 641-792, 1991. [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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