臺灣博碩士論文加值系統

我們研究一些疊代法(如Lanczos，GMRES及他們的各種變形)和區域分割法來求解非線性橢圓特徵值問題。首先，我們利用Lanczos方法解多重右邊向量的線性系統。我們給出求第二組線性系統近似解的一個誤差上限，其中使用Lanczos-Galerkin程序解第一組線性系統。我們也探索了所提出的數值方法對延續問題的可能應用。接著，對Lanczos方法的一種變形MINRES，我們給了一些收斂定理。Multigrid-Lanczos及Multigrid-GMRES算則被提出來求解對應的離散問題的分支解及偵測分支解的奇異點。這些算則有健全且易執行的優點。最後，我們使用非重疊及重疊的區域分割法求解四階非線性特徵值問題。對線性的方程，我們利用Fourier分析及探針技巧對介面系統找出前置矩陣，方法類似於Chan等人的推導。我們的數值結果證實這些算則的效力。

We study some iterative methods (such as the Lanczos method, GMRES and their other variants) and domain decomposition methods for solving nonlinear elliptic eigenvalue problems. First, we apply the Lanczos method to solve linear systems with multiple right hand side. We give an error bound for the approximate solution of the second linear system, where the Lanczos-Galerkin process is used to solve the first linear system. We also seek the possible application of the proposed numerical method to continuation problems. Next, we show some convergence theory for the MINRES, a variant of the Lanczos method. A multigrid-Lanczos and a multigrid-GMRES algorithm are proposed for tracking solution branches of associated discrete problem and detecting singular points along solution branches. The proposed algorithms have the advantage of being robust and easy to implement. Finally, we show how nonoverlapping and overlapping domain decomposition methods can be used to solve fourth order nonlinear elliptic eigenvalue problems. For the linearized von Kármán equation, we present preconditioners using both Fourier analysis and probing techniques for the interface systems, which are similar to those derived by Chan et al. Our numerical results show the efficiency of these algorithms.

1. Introduction ---------------------------------------------------------1
2. Iterative methods and multigrid algorithms --------------------6
2.1 The Lanczos algorithms -------------------------------------------------------7
2.1.1 A brief review of well-known algorithms -----------------------------7
2.1.2 The restarted Lanczos-Galerkin method -----------------------------11
2.1.3 The preconditioned Lanczos algorithm and iterative refinement -13
2.1.4 The block Lanczos method --------------------------------------------15
2.1.5 Application to continuation problems --------------------------------15
2.2 A multigrid-Lanczos algorithm ----------------------------------------18
2.2.1 Some convergence theory for the Lanczos Method ----------------18
2.2.2 A multigrid-Lanczos algorithm ---------------------------------------21
2.3 GMRES algorithms ---------------------------------------------------------24
2.3.1 GMRES and preconditioned GMRES -------------------------------24
2.3.2 Block GMRES and preconditioned block GMRES ----------------26
2.4 A multigrid-GMRES algorithm -------------------------------------------30
3. Domain Decomposition methods for fourth order nonlinear elliptic
eigenvalue problems------------------------------------------------------------32
3.1 A brief review of domain decomposition for linear problems ----------32
3.1.1 Fourier analysis of the interface operator ----------------------------34
3.1.2 The interface probing preconditioner --------------------------------39
3.2 Nonoverlapping domain decomposition for nonlinear problems ------39
3.2.1 Discretization of the von Kármán equations on the
nonoverlapping domain -----------------------------------------40
3.2.2 The interface probing preconditioner for nonlinear systems ------42
3.3 Overlapping domain decomposition for nonlinear problems ----------43
3.3.1 Discretization of the von Kármán equations on the overlapping
domain --------------------------------------------------------------------44
3.3.2 The Schwarz alternating procedures ---------------------------------47
4. Numerical results -------------------------------------------------------------50
List of figures ----------------------------------------------------------------60
List of tables ---------------------------------------------------------------71
Appendix A ----------------------------------------------------------83
References ------------------------------------------------------------------87