# 臺灣博碩士論文加值系統

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 本論文主旨是針對橢圓型方程尋求其數值解，其中所採用的數值方法為Trefftz以及collocation 法。整個論文分成兩個部分：第一部份介紹邊界型的collocation Trefftz 法；我們用此方法解決帶奇異性的Poisson方程及biharmonic方程，提供演算法並做了誤差分析。在第二部份則是探討區域型collocation法以及將它與有限元法相結合；用此兩類方法來解決帶奇異性的Poisson方程，提供具體的結合架構也做了誤差分析。在整個論文的分析過程中有一個重要且特殊的現象。即數值積分公式逼近積分項時，採用的數值積分公式僅影響一致橢圓不等式（強制性），並不影響整體誤差（最優估計）。因此在collocation Trefftz 或collocation法中我們可採用Gaussian積分公式的積分點作為collocation點，也可以採用Newton-Cotes積分公式的積分點作為collocation點。只要collocation點的數目夠多且夠密集，即可滿足一致橢圓不等式。基於這樣的作法及分析方式，我們的數值方法可以更廣泛被使用，並不局限於簡單矩形區域而已。對於一般多邊形區域問題，可將整個區域分成數個子區域，每個子區域可使用不同的基函數、不同的項數去逼近，以達最好的效能。當然也可搭配其它類型的數值方法（有限元素法或是有限差分法），來解決更複雜問題。相較於現存相關文獻的作法及分析，本論文內所提供的演算法及分析方式是更具彈性的。在每章末節皆有數值結果來驗證我們所提出的理論分析，結果相吻合。
 The dissertation consists of two parts.The first part is mainly to provide the algorithms and error estimates of the collocation Trefftz methods (CTMs) for seeking the solutions of partial differential equations. We consider several popular models of PDEs with singularities, including Poisson equations and the biharmonic equations. The second part is to present the collocation methods (CMs) and to give a unified framework of combinations of CMs with other numerical methods such as finite element method, etc. An interesting fact has been justified: The integration quadrature formulas only affect on the uniformly \$V_h\$-elliptic inequality, not on the solution accuracy. In CTMs and CMs, the Gaussian quadrature points will be chosen as the collocation points. Of course, the Newton-Cotes quadrature points can be applied as well. We need a suitable dense points to guarantee the uniformly \$V_h\$-elliptic inequality. In addition, the solution domain of problems may not be confined in polygons. We may also divide the domain into several small subdomains. For the smooth solutions of problems, the different degree polynomials can be chosen to approximate the solutions properly. However, different kinds of admissible functions may also be used in the methods given in this dissertation. Besides, a new unified framework of combinations of CMs with other methods will be analyzed. In this dissertation, the new analysis is more flexible towards the practical problems and is easy to fit into rather arbitrary domains. Thus is a great distinctive feature from that in the existing literatures of CTMs and CMs. Finally, a few numerical experiments for smooth and singularity problems are provided to display effectiveness of the methods proposed, and to support the analysis made.
 CHAPTER 0. OverviewCHAPTER 1. The CTM for Motz''s and Cracked Beam ProblemsCHAPTER 2. The CTM for Biharmonic EquationsCHAPTER 3. Collocation MethodsCHAPTER 4. Combinations of CM and FEMCHAPTER 5. Radial Basis Collocation Methods