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研究生:林雅菁
研究生(外文):Ya Jing Lin
論文名稱:使用多重網格有限元素方法和雙重網格有限差分法解非線性特徵值問題
論文名稱(外文):Multigrid and two-grid discretization schemes for nonlinear eigenvalue problems
指導教授:簡澄陞
指導教授(外文):Cheng-Sheng Chien
學位類別:碩士
校院名稱:國立中興大學
系所名稱:應用數學系
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2004
畢業學年度:92
語文別:中文
論文頁數:28
中文關鍵詞:非線性橢圓特徵值問題分歧點延續法多重網格法雙重網格法Lanczos 方法MINRESSYMMLQ
外文關鍵詞:nonlinear elliptic eigenvalue problemsbifurcation pointscontinuation methodsmultigrid methodstwo-grid schemesLanczos methodsMINRESSYMMLQ
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在此篇論文裡,我們研究多重網格法應用在延續法中,來解非線性橢圓特徵值問題,並且討論利用雙重網格法來解定義在L型曲域的線性特徵值問題。我們首先用有限元素法將偏微分方程離散化,在多重網格法的V-循環,W-循環以及完全的多重網格V-循環中,使用Lanczos法、MINRES和SYMMLQ作為其中的鬆弛法。我們比較這些鬆弛法在多重網格中的效益。我們亦討論雙重網格-中央差分法在L型的區域中解線性特徵值問題。由數值的結果中我們可以知道 Lanczos法是相當有效的。最後我們將所得結果繪製成圖表並做結論。

We study multigrid methods in the context of continuation methods for semilinear elliptic eigenvalue problems, where the Lanczos method, the MINRES and the SYMMLQ are used as linear solvers.
The semilinear elliptic eigenvalue problems are discretized by six-node triangular elements.
We compare the efficiency of these linear solvers in the context of multigrid V-cycle, W-cycle and the full multigrid V-cycle (FMG) schemes.
Next, we study the two-grid centered difference discretization scheme for the eigenvalue problem defined on an L-shaped domain.
Our numerical results show that the accuracy of computed eigenpairs is improved efficiently.
Moreover, compared with the performance of (preconditioned) MINRES and SYMMLQ, the (preconditioned) Lanczos algorithm is still very competitive.

Contents
1.Introduction 1
2.A brief review of the Preconditioned Lanczos type algorithms 2
3.The finite-element method 7
4.V-cycle, W-cycle and full multigrid V-cycle methods 10
5.Two-grid discretization schemes and domain decomposition 13
6.Numerical results 16
7.Conclusions 18
List of tables 19
List of figures 24
References 27

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