在文獻中，邊界近似法(BAM)，又稱Collocation Trefftz Method，是一種有效解決具有奇異點橢圓邊值問題的方法。在實際的計算過程中BAM包含很多種類，其中包括數值邊界近似法及符號邊界近似法和它們的變形，而符號邊界近似法的演算過程時間通常是較數值邊界近似法還要慢。在本篇論文中將改善符號邊界近似法，使此種方法成為所有BAM中演算過程最快速的方法。我們在此篇論文中證明了一些重要的引理來減少計算時間，而且找到遞迴式以便於加速計算積分過程。在符號邊界近似法中，它的缺點則為condition number非常大。在此我們找到了一個好的而且簡單的preconditioner來減少condition number，且分別對Motz problem和Schiff’s Model兩種模型來做一些數值結果與比較。

Boundary Approximation Method (BAM), or the Collocation Trefftz Method called in the literature, is the most efficient method to solve elliptic boundary value problems with singularities. There are several versions of BAM in practical computation, including the Numerical BAM, Symbolic BAM and their variants. It is known that the Symbolic BAM is much slower than Numerical counterpart. In this thesis, we improve the Symbolic BAM to become the fastest method among all versions of BAM. We prove several important lemmas to reduce the computing time, and a recursive procedure is found to expedite the evaluation of major integrals. Another drawback of the Symbolic BAM is its large condition number. We find a good and easy preconditioner to significantly reduce the condition number. The numerical experiments and comparison are also provided for the Motz problem, a prototype of Laplace boundary value problem with singularity, and the Schiff's Model, a prototype of biharmonic boundary value problem with singularity.

1.Introduction..........................3
2.Boundary Approximation Method.........5
3.Motz Problem.........................10
4.Preconditioning......................28
5.Schiff's Model.......................35

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