臺灣博碩士論文加值系統

本論文中將探討來自於偏微分方程經離散化後所得之非線性特徵值問題。吾人結合多重網格法求解線性系統，以及網格轉換法改善逆迭代法中之初始猜測向量等技巧，提出多重網格形態之逆迭代法。並且將上述非線性特徵值問題經矩陣轉換後，應用此方法求解絕對最小之特徵值。此方法最後將應用於探討挫屈問題之穩定性。

The article is concerned with aspects of the nonlinear eigenvalue problems which arise from partial differential equation by finite difference discretization: T(λ)x=0, where
T(λ) is a n ×n matrix whose elements are analytical functions in parameter λ. We shall propose the multigrid type inverse
iteration algorithms which use the multigrid linear solver for
finding the smallest eigenvalue in magnitude of the nonlinear
eigenvalue problem and grid transform strategy for finding the
initial vector. The methods finally illustrate by numerical
results from experiments with buckling problem.

1 Introduction
2 Multigrid Methods
2.1 Basic Iterative Methods
2.1.1 Convergence of Iterative Method
2.1.2 The Jacobi Method
2.1.3 Weighted Jacobi Method
2.2.4 The Gauss-Seidel Method
2.2 General Two-Grid Method
2.2.1 Motivation and Numerical Observation
2.2.2 Two-Grid Algorithm
2.2.3 Restriction
2.2.4 Prolongation
2.3 A Complete Multigrid Method
2.3.1 Multigrid μ-cycle Scheme
2.3.2 Full Multigrid V-cycle Schemes
3 Inverse Iteration And Multigrid Inverse Iteration
3.1 Power Method
3.2 Inverse Iteration
3.3 Multigrid Inverse Iteration
3.4 Nonlinear Eigenvalue Problem
4 Application And Numerical Experiments
4.1 A Simple Experiments
4.2 Boundary Value Eigenvalue Problem
4.3 Buckling Problem
5 Conclusion

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