Please use this identifier to cite or link to this item: http://hdl.handle.net/11455/18133
標題: 雙重網格法處理特徵值問題
Two Grid Discretization Schemes for Eigenvalue Problems
作者: 鄭博文
Jeng, Bor-Wen
關鍵字: 雙重網格法
two-grid scheme
非線性特徵值問題
線性特徵值問題
薛丁格特徵值問題
薛丁格-波松系統
延續法
有限元素法
有限差分法
奇異點
週期性邊界條件
nonlinear eigenvalue problem
linear eigenvalue problem
Schrdinger eigenvalue problem
Schrdinger-Poisson system
continuation method
finite element method
finite difference method
singular points
periodic boundary conditions
出版社: 應用數學系
摘要: 在這博士論文中,我們研究一些高效率的雙重網格離散法來求解特徵值問題,包括半線性橢圓特徵值問題、線性特徵值問題、Schrödinger特徵值問題、以及非線性Schrödinger-Poisson特徵值問題。首先,我們提出具有雙重迴圈延續法的雙重網格有限元素和有限差分離散法來追蹤半線性橢圓特徵值問題的解分支。我們也將對稱簡化的技巧與雙重網格有限元素離散法做結合來追蹤這些解曲線。廣泛的數值試驗顯示對求解二階半線性橢圓特徵值問題來說,我們所提出的具有雙重迴圈延續法的雙重網格離散法是很有效率且強健的方法。 其次,我們提出一個新的雙重網格中央差分離散法來計算具有Dirichlet或週期性邊界條件的自伴隨偏微分運算子的一些極小特徵對偶,此問題的特徵值一般來說會具有重複和聚集的特性。奠基在此方法上,我們更進一步地發展出一個新的雙重網格中央差分法來處理非線性Schrödinger-Poisson特徵值問題的數值解。我們的數值結果顯示出在Schrödinger-Poisson系統中所考慮的摻雜劑如何影響Schrödinger特徵值問題的最初的幾個特徵對偶。 最後,考慮定義在矩形區域具有週期性邊界條件的二階線性橢圓特徵值問題,我們探討相對應於此問題的變分形式的正確特徵對偶的存在性。然後,我們描述一個雙重網格有限元離散法來計算具有週期性邊界條件的Schrödinger特徵值問題的前幾個極小特徵對偶。六節點的三角形元素被利用來離散化我們的問題。我們的數值結果顯示出此演算法具有O(h^4)的收斂速率。此外,從Schrödinger特徵值問題的數值計算上也證實了此雙重網格法的效力和強健性。
In this dissertation we study some efficient two-grid discretization schemes for solving eigenvalue problems, including semilinear elliptic eigenvalue problems, linear eigenvalue problems, Schrdinger eigenvalue problems, and nonlinear Schrdinger-Poisson eigenvalue problems. First, we present two-grid finite element and finite difference discretization schemes with two-loop continuation algorithm for tracing solution branches of semilinear elliptic eigenvalue problems. We also combine the symmetry reduction technique with the two-grid finite element discretization scheme to trace the solution curves. Comprehensive numerical experiments show that the two-grid discretization schemes with two-loop continuation algorithm are efficient and robust for solving second order semilinear elliptic eigenvalue problems. Next, we present a new implementation of the two-grid centered difference discretization scheme for computing extremum eigenpairs of self-adjoint partial differential operators with Dirichlet or periodic boundary conditions, which in general possess multiple and clustered eigenvalues. Based on this method we develop a novel two-grid centered difference method for the numerical solutions of the nonlinear Schrdinger-Poisson eigenvalue problem. Our numerical results show how the first few eigenpairs of the Schrdinger eigenvalue problem are affected by the dopant which is considered in the Schrdinger-Poisson system. Finally, we study the existence of exact eigenpairs of the corresponding variational formulation to the second-order linear elliptic eigenvalue problems on a rectangular domain with periodic boundary conditions. Then we describe a two-grid finite element discretization scheme for finding the extremum eigenpairs of the Schrdinger eigenvalue problem with periodic boundary conditions. The six-node triangular elements were exploited to discretize the PDEs. Our numerical results show that the rate of convergence of the proposed algorithm is O(h^4). Moreover, numerical implementations on the Schrdinger eigenvalue problem show the efficiency and robustness of the two-grid scheme we propose.
URI: http://hdl.handle.net/11455/18133
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