臺灣博碩士論文加值系統

基本解法(method of fundamental solutions)與Trefftz法都試指數收斂的數值方法，換言之，誤差的對數值正比於空間離散化中的節點數目或Tefftz的基底數。首先，我們應用基本解法來求解拉普拉斯(Laplace)方程式，我們的解答呈現指數收斂的特性。我們考慮以下幾何圖形：矩形、橢圓、阿米巴和長方體等形狀。在解決的過程中，基本解法的源點增加會有較良好的準度，但是會造成不穩定的系統矩陣而使用MPFR(multiple precision floating-point reliable)庫來避免以上不穩定的系統矩陣。初步結果顯示，在面積/周長比率較大時，收斂速度會更快。另一方面，本研究就是突破IEEE 754規範中所限制的單精度、雙精度之規範，讓使用者能自行帶入尾數，加以延伸，來實現以往不可能達到的精度計算。並且本研究將和解析解來比較，借此印證本研究之高精度計算的準確度。
並且我們發現用LU分解法來解Trefftz法的系統矩陣相對比較穩定，並且導出一個範圍公式可以推估其特徵長度，我們也用MPFR庫來驗證以上的公式，來得到更為理想的答案，並且驗證其指數收斂。

It is well known that both the method of fundamental solutions (MFS) and the Trefftz method are numerical methods of exponential convergence. In other words, the logarithmic error is proportional to the node number of spatial discretization or the number of the Trefftz basis. In this study, the exponential convergence of MFS is demonstrated by solving Laplace equation in domains of rectangles, ellipses, amoeba-like shapes, and rectangular cuboids. In the solution procedure, the sources of the MFS are located as far as possible and the instability resulted from the ill-conditioning of system matrix is avoided by using the multiple precision floating-point reliable (MPFR) library. The results converge faster for the cases of smoother boundary conditions and larger area/perimeter ratios. For the Trefftz method, it is found that numerical solutions obtained the LU decomposition is more stable. In addition, a formula is derived to predict the feasible range of the Trefftz characteristic length. The MPFR library is used to validate the range formula and to demonstrate the exponential convergence of the Trefftz method.

目錄

摘要 I
Abstract II
致謝 III
表目錄 IV
圖目錄 V
符號說明VI
第一章 前言
1.1研究動機
1.2研究目的
1.3本文組織
第二章 文獻回顧
2.1 高精度計算
2.2 基本解法與回顧
2.3 修正型Trefftz法(MCTM)
第三章 數值模式
3.1 基本解法
3.1.1 源點的放置法
3.2 增強型基本解法
3.3 Trefftz法
3.4 修正型Trefftz 法(MCTM)
3.4.1浮點數運算和其可接受的特徵長度
3.5預測特徵長度
第四章數值結果
4.1 MFS計算結果
4.1.1 三維方塊類型
4.1.2 不連續角類型
4.1.3非調和函數類型
4.1.4 MFS指數收斂
4.2修正型Trefftz法(MCTM)計算結果
4.2.1 外旋輪線(Epitrochoid)
4.2.2橢圓(Ellipse)類型
4.2.3阿米巴類型
4.2.4柯西問題類型
4.2.5 MCTM指數收斂
4.3時間效率
4.4 MCTM和MFS之比較
第五章結果討論
參考文獻
附錄

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