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研究生:沈鴻洲
研究生(外文):Hung-Jou Shen
論文名稱:用非協調有限元求解特徵值問題
論文名稱(外文):Non-conforming Finite Element Methods for Eigenvalue Problems
指導教授:李子才
指導教授(外文):Zi-Cai Li
學位類別:碩士
校院名稱:國立中山大學
系所名稱:應用數學系研究所
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:2005
畢業學年度:93
語文別:英文
論文頁數:50
中文關鍵詞: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
外文關鍵詞:extension of rotated bilinear elementrotated bilinear elementWilson''s elementbilinear elementlinear elementnon- conformingeigenvalueconforming
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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。
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)
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
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