邊界近似法在工程文獻上亦稱為 Collocation Trefftz Method，被用來求
解在矩形域下的 Laplace 邊值問題。假定對於整個定義域選擇一特解，如
果在其他頂點上沒有奇異性，則此方法所產生的誤差應該為指數收斂。否
則，它將會退化成多項式收斂，這個收斂的階與奇異性的強度有著某種的
關係。因此，我們可以很容易地設計出帶有某種收斂階的模型。
在一扇形的定義域上，當一側的邊界條件為一超越函數時，它必須藉
由冪級數去逼近。當我們在多邊形上求解 Laplace 方程式時，冪級數的截
斷將產生人為的奇異性，因而它將減慢我們期望的收斂階。這篇論文中，
我們研究截斷誤差是如何影響我們的收斂速度。此外，我們探討從一收斂
階到另一階的轉變行為。最後，我們也討論在邊界近似法上使用增廣基底
的效果。

Boundary approximation method, also known as the collocation Trefftz method in
engineering, is used to solve Laplace boundary value problem on rectanglular domain.
Suppose the particular solutions are chosen for the whole domain. If there is no singularity
on other vertices, it should have exponential convergence. Otherwise, it will
degenerate to polynomial convergence. In the latter case, the order of convergence has
some relation with the intensity of singularity. So, it is easy to design models with
desired convergent orders.
On a sectorial domain, when one side of the boundary conditions is a transcendental
function, it needs to be approximated by power series. The truncation of this power
series will generate an artificial singularity when solving Laplace equation on polygon.
So it will greatly slow down the expected order of convergence. This thesis study how
the truncation error affects the convergent speed. Moreover, we focus on the transition
behavior of the convergence from one order to another. In the end, we also apply our
results to boundary approximation method with enriched basis.

1 Introduction 2
2 Solutions for Laplace Equation 3
2.1 Solution for D-D type . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Solution for N-N type . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Solution for N-D type . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Solution for D-N type . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.5 Analysis for Singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Boundary Approximation Method 13
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Convergent Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4 Singularity and Convergent Order 20
5 Transition of Convergent Order 25
5.1 Sine Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.2 Cosine Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6 Enriched Basis Functions 38
6.1 Enriched at Vertex A . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
6.2 Enriched at Vertex D . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
7 Conclusion 52

[1] Z. C. Li, T. T. Lu, H. Y. Hu and A. H. D. Cheng, Particular solutions of Laplace’s
equations on polygons and new models involving mild singularities, Engineering
Analysis with Boundary elements, Vol. 29, pp. 59-75, 2005.
[2] Z. C. Li, T. T. Lu, H. Y. Hu and A. H. D. Cheng, Trefftz and Collocation Methods,
WIT, Southsampton, Boston, 2008.
[3] C. H. Chen, Further study on Motz problem, Master Thesis, National Sun Yat-sen
University, 1998.
[4] L. T. Tang, Cracked-Beam and Related Singularity Problems, Master Thesis, National
Sun Yat-sen University, 2001.
[5] J. R. Wang, Convergence Analysis of BAM on Laplace BVP with Singularities,
Master Thesis, National Sun Yat-sen University, 2006.
[6] Carlos J.S. Alves and Vitor M.A Leit˜ao, Crack analysis using an enriched MFS
domain decomposition technique, Engineering Analysis with Boundary Elements,
pp. 160-166, 2006.
[7] Bernal F and Kindelan M. On the enriched RBF method for singular potential
problems, Engineering Analysis with Boundary Elements, 2009.