Applications of Domain Decomposition Method and the Method of Fundamental Solutions for Some Partial Differential Equations with Clustering

本論文主要在探討區域分割法以及無網格基本解法的結合，並應用於求解拉普拉斯方程式、史托克斯方程式、擴散方程式以及對流-擴散方程式的成果。基本解法是一種無網格且具高效率計算的數值方法，但此種方法，在計算點數越多的情況下，會導致基本解所構成的線性系統矩陣越趨於病態。此問題在基本解法的發展上，造成對於計算規模的限制。因此，本論文提出此建立在區域分割理論上的基本解法，來解決這個病態矩陣的問題。應用此種方法，本文針對尖角計算域、穩態史托克斯流場、以及跟時間等相關的問題，做一系列的探討。值得注意的是，對於尖角計算域問題的計算，本篇論文是先應用保角轉換，使尖角計算域轉換為半無窮計算域，再將此區域分割並用基本解法來求解。由於區域分割法具有應用在平行計算的潛力，因此，本文也將此方法應用於平行計算上之效率討論。This thesis mainly describes the combination of the Domain Decomposition Method (DDM) and the Method of Fundamental Solutions (MFS) as a meshless numerical method (DDM-MFS) to solve problems governed by various Partial Differential Equations (PDEs), including Laplace equation, Stokes’ equations, diffusion equation and advection-diffusion equation. The MFS is an efficient meshless method, which its application can be extensively found in the literature. However, the resultant matrix in the MFS will approach to ill-conditioned as computational points increasing, which creates restrictions for development of this method. The DDM-MFS is thus developed in order to solve this problem. Applied the DDM-MFS, problems including the cusp domain problems, the steady Stokes’ flow problems and the time-dependent problems are successfully addressed. It is to be noted that there is a new approach to solve the cusp problems proposed in this thesis, which is to transform the cusp domains in to semi-infinite domains by conformal mapping techniques. Since the DDM is developed as a preparation for parallel computing, the efficiency of parallel computing applied the DDM-MFS is also tested for the problems mentioned above in the thesis.誌謝 (i) 摘要 (ii) Abstract (iii) Table of Contents (iv) List of Figures (vi) List of Tables (x) Chapter 1 Introduction (1) 1.1 Motivation and objectives (1) 1.1.1 Mesh dependent numerical methods (2) 1.1.2 Meshless numerical methods (3) 1.1.3 The method of fundamental solutions (5) 1.1.4 Domain decomposition methods (6) 1.1.5 Objectives (7) 1.2 Organization of the thesis (7) Chapter 2 Formulation of the DDM-MFS and its robustness test (12) 2.1 Introduction (12) 2.2 Formulations of the DDM-MFS (12) 2.3 Robustness test of the DDM-MFS (18) Chapter 3 Applications of the DDM-MFS (31) 3.1 Introduction (31) 3.2 Cusp domain problems (33) 3.3 Stokes flow problems (38) 3.4 Time- dependent problems (44) 3.4.1 The DDM-MFS for the diffusion equation (44) 3.4.2 The DDM-MFS for the advection-diffusion equation (47) Chapter 4 Parallel computing applied the DDM-MFS (72) 4.1 Introduction (72) 4.2 Procedures of parallelization (72) 4.3 Efficiency analysis (76) Chapter 5 Conclusions and future works (80) 5.1 Conclusions (80) 5.2 Future works (81) Appendix A Derivations for Cusp Domain Problems (82) 簡歷 (91