利用配置Trefftz方法（CTM）解環狀區域問題，在誤差分析上產生最佳的收斂速率。這方法普遍應用於特殊例子：在狄利克雷問題解環形域含圓洞的環狀模型，並比較零場法（NFM）與內場法（IFM），而我們發現誤差和條件數都很小。
在這論文中零場法是利用基本解將外節點代入Green公式所解環狀區域的解。利用傅立葉展開式來選擇狄利克雷邊界條件或諾依曼邊界條件當作已知或未知條件解環狀邊界，這樣顯式離散方程可以得到正交性傅立葉函數。拉普拉斯方程中零場法也應用於多洞的橢圓方程和特徵值問題，而我們也探討多洞的環狀問題。對於第一類邊界積分方程（BIE）用三角函數誤差分析做無限解，而得出指數收斂速率。
事實上，配置Trefftz方法不僅可以適用於任意的模型，而且有好的數值結果來驗證而且有更好的數值結果。由於配置Trefftz方法的演算法簡單，編程容易，建議用配置法解環狀含有圓洞的工程問題。

The collocation Trefftz method (CTM) proposed in [36] is employed to annular shaped domains, and new error analysis is made to yield the optimal convergence rates. This popular method is then applied to the special case: the Dirichlet problems on circular domains with circular holes, and comparisons are made with the null field method (NFM) proposed , and new interior field method (IFM) proposed in [35], to find out that both
errors and condition numbers are smaller.

Recently, for circular domains with circular holes, the null fields method (NFM) is proposed by Chen and his groups. In NFM, the fundamental solutions (FS) with the source nodes Q outside of the solution domains are used in the Green formulas, and the FS are replaced by their series expansions. The Fourier expansions of the known or the unknown Dirichlet and Neumann boundary conditions on the circular boundaries are chosen, so that the explicit discrete equations can be easily obtained by means of orthogonality of Fourier functions. The NFM has been applied to elliptic equations and eigenvalue problems in circular domains with multiple holes, reported in many papers; here we cite those for Laplace’s equation only (see [18, 19, 20]). For the boundary integral equation (BIE) of the first kind, the trigonometric functions are used in Arnold [4, 5], and error analysis is made for infinite smooth solutions, to derive the exponential convergence rates. In Cheng’s Dissertation [21, 22], for BIE of the first kind, the source nodes are
located outside of the solution domain, the linear combination of fundamental solutions are used, and error analysis is made only for circular domains.

This fact implies that not only can the CTM be applied to arbitrary domains, but also a better numerical performance is provided. Since the algorithms of the CTM is simple and its programming is easy, the CTM is strongly recommended to replace the NFM for circular domains with circular holes in engineering problems.

Abstract i
1 The Collocation Trefftz Method for Laplace’s Equation 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Collocation Trefftz Method . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Basic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.2 Model Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Relations of the CTM to Other Methods . . . . . . . . . . . . . . . . . . 6
1.3.1 The Null Field Method . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.2 The Interior Field Method . . . . . . . . . . . . . . . . . . . . . . 11
1.3.3 The First Kind Boundary Integral Equations . . . . . . . . . . . . 11
1.3.4 Relations of TM with IFM and the first kind BIE . . . . . . . . . 12
1.4 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4.1 Preliminary lemmas . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4.2 Error Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.3 Error Bounds for Annular Shaped Domains . . . . . . . . . . . . 21
1.5 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.5.1 Definitions of Boundary Errors . . . . . . . . . . . . . . . . . . . 22
1.5.2 Model Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.5.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.5.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 28
2 Different Methods for Multiple Circular Holes 35
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2 Explicit Algorithms of NFM for Multiple Circular Holes . . . . . . . . . . 36
2.2.1 Basic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2.2 Explicit Algebraic Equations . . . . . . . . . . . . . . . . . . . . 38
2.2.3 Explicit Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.3 The Algorithms of the IFM for Two Holes . . . . . . . . . . . . . . . . . 47
2.4 The Explicit Equations of NFM for Multiple Holes . . . . . . . . . . . . 49
2.5 Simple Algorithms of IFM and CTM . . . . . . . . . . . . . . . . . . . . 52
2.6 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.6.1 The NFM for Circular Domain with Multiple Circular Holes . . . 52
2.6.2 The IFM for Circular Domain with Multiple Circular Holes . . . . 54
2.6.3 The CTM for Circular Domain with Multiple Circular Holes . . . 55
2.6.4 Simplified Equations of NFM and IFM . . . . . . . . . . . . . . . 56
2.6.5 Relations of CTM and IFM . . . . . . . . . . . . . . . . . . . . . 57
2.7 Conditions of Normal Derivations . . . . . . . . . . . . . . . . . . . . . . 58
2.8 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.8.1 Computations by the NFM and CTM . . . . . . . . . . . . . . . . 61
2.8.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3 Elliptic Domains by the Null Field Method 77
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.2 Elliptic Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.2.2 Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . 80
3.2.3 Derivatives and Integrations under Elliptic Coordinates . . . . . . 82
3.3 The Null Field Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.3.1 Basic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.3.2 Explicit Algebraic Equations . . . . . . . . . . . . . . . . . . . . 88
3.3.3 Explicit Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.3.4 The degenerate Case of @SR1 to a Strip . . . . . . . . . . . . . . . 95
3.4 The Algorithms of the Interior Field Method . . . . . . . . . . . . . . . . 96
3.5 The Collocation Trefftz Method . . . . . . . . . . . . . . . . . . . . . . . 97
3.6 The Explicit Equations of the NFM for Two Holes . . . . . . . . . . . . . 98
3.7 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
3.7.1 Model Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
3.7.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . 105

[1] H. Ammari, H. Kang, G. Nakamura and K. Tanuma, Complete asymptotic expansions of solutions
of the system of elastostatics in presence of an inclusion of small diameter and detection of an
inclusion, J. Of elasticity, Vol. 67, pp. 97 – 129, 2002.
[2] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas,
Graphs and Mathematical Tables, Dover Publications, Inc. New York, 1964.
[3] W. T. Ang and I. Kang, A complex variable boundary element method for elliptic partial differential
equations in a multiple-connected region, Inter. J. Computer Math., Vol. 75, pp. 515 – 525, 2000.
[4] D. N. Arnold, A spline-trigonometric Galerkin method and an exponentially convergent boundary
integral method, Math. Comp., Vol. 41, pp. 383 - 397, 1983.
[5] D. N. Arnold and W. L. Wendland, On the asymptotic convergence of collocation methods, Math.
Comp., Vol. 41, pp. 349 - 381, 1983.
[6] K. E. Atkinon, A Survey of Numerical Methods for the Solutions of Fredholm Integral
Equations of the Second Kind, Cambridge University Press, 1997.
[7] M. R. Barone and D. A. Caulk, Special boundary integral equations for approximate solution of
Laplace’s equation in two-dimensional regions with circular holes, Q. Ji Mech. appl. Math, Vol. 34.
pp. 265 – 286, 1981.
[8] M. R. Barone and D. A. Caulk, Special boundary integral equations for approximate solution of
potential problem in three-dimensional regions with slender cavities of circular cross-section, IMA
J. Appl. Math., Vol. 35, pp. 311 – 325, 1985.
[9] M. D. Bird and C. R. Steele, A solution procedure for Laplace’s equation on multiply
connected circular domains, Transaction ASME, Vol. 59, pp. 398 –404, 1992.
[10] C. A. Brebbia, J.C.F. Telles and L.C. Wrobel, Boundary Element Techniques, Theory and
Applications in Engineering, Springer-Verlag, Berlin Heidelberg, New York, 1984.
[11] D. A. Caulk, Analysis of steady heat conduction in regions with circular holes by a special boundary-
integral method, IMA J. Appl. Math., Vol. 30, pp. 231 – 246, 1983.
[12] C. Canuto and A. Quarteroni, Approximation results for orthogonal polynomial in Sobolev spaces,
Math. Comp, Vol. 38, pp. 67 – 86, 1982.
[13] G. Chen and J. Zhou, Boundary Element Methods, Academic Press, Chapter 9, New York,
1992.
[14] J. T. Chen, C.T. Chen, P.Y. Chen and I. L. Chen, A semi-analytical approach for radiation and scat-
tering problem with circular boundaries, Computer Methods in Applied Mechanics and Engineering,
Vol. 196, pp. 2751 – 2764, 2007.
[15] J. T. Chen, Y. T. Lee. and J. W. Lee, Torsional rigidity of an elliptic bar with multiple elliptic
inclusions using the null-field integral approach, To appear in Comput Mech.
[16] J. T. Chen, S. K. Kao, W. M. Lee and Y. T. Lee, Eigensolutions of the Helmholtz equation for a
multiply connected domain with circular boundaries using the multipole Trefftz method, Engineering
Analysis with Boundary Elements, Vol. 34, pp. 463 – 470, 2010.
[17] J. T. Chen, J. H. Lin, S. R. Kuo and Y. P. Chui, Analytical study and numerical experiments for
degenerate scale problems in the boundary element method using degenerate kernel and circulants,
Engineering Analysis with Boundary Elements, Vol. 25, pp. 819 – 828, 2001.
[18] J. T. Chen and W. C. Shen, Null-field approach for Laplace problems with circular boundaries using
degenerate kernels, Numer. Meth. PDE., Vol. 25, pp. 63 – 86, 2009.
[19] J. T. Chen, S. R. Kuo and J. H. Lin, Analytical study and numerical experiments for degenerate
scale problems in the boundary element method for two-dimensional elasticity, Engineering Analysis
with Boundary Elements, Vol. 26, pp. 1669 – 1681, 2002.
[20] J. T. Chen, C. F. Lee. J. L. Chen and J. H. Lin, An alternative method for degenerate scale
problem in boundary element methods for two-dimensional Laplace equation, Engineering Analysis
with Boundary Elements, Vol. 26, pp. 559 – 569, 2002.
[21] R. S. C. Cheng, The delta-trigonometric and spline-trigonometric methods using the single layer
potential representation, Ph. Dissertation, University of Maryland, 1987.
[22] R. S. C. Cheng and D. N. Arnold, The delta-trigonometric using the single layer potential repre-
sentation, J. of Integral Equations and Applications, Vol. 1, pp. 517 –547, 1988.
[23] P. F. Davis. and P. Rabinowitz, Methods of Numerical Integral Integration(Sec. Ed.), Academic
Press, Inc, New York, 1983.
[24] I. S. Gradsheyan. and Z. M. Ryzhik, Tables of Integrals, Series and Products, Academic
Press, New York, 1965.
[25] G. C. Hsia and W. L. Wendland, Boundary Integral Equations, Springer, Berlin, 2008.
[26] H. O. Kreiss and J. Oliger, Stability of the Fourier method, SIAM J. Numer. Anal., Vol, 16, pp. 421
– 433, 1979.
[27] Z.C. Li, Combined Methods for Elliptic Equations with Singularities, Interfaces and
Infinities, Kluwer Academic Publishers, Boston 1998.
[28] Z.C. Li, Method of fundamental solutions for annular shaped domains, J. Comp. and Appl. Math.,
vol. 228, pp. 355 - 372, 2009.
[29] Z. C. Li, C. S. Chien and H. T. Huang, Effective Condition number for finite difference method, J.
Computational and Applied Mathematics, Vol. 198, pp. 208-235, 2007.
[30] Z. C. Li, J. Huang and H. T. Huang, Stability analysis of method of fundamental solutions for mixed
boundary value problems of Laplace’s equation, Computing, Vol. 88, pp. 1 – 29, 2010.
[31] Z.C. Li, M. G. Lee, H.T. Huang and J.T. Chen, Error analysis of the null-field method for Dirichlet
problem of Laplace’s equation on circular domains with circlar holes, manuscript, 2010.
[32] Z. C. Li, C. P. Liaw, H. T. Huang and M. G. Lee, The null-field method of Dirichlet problems
Laplace’s equation on circular domains with circular holes, Technical report, National Sun Yat-sen
University, Kaohsiung, Taiwan, 2010.
[33] Z.C. Li, C. P. Liaw, M. G. Lee and H. T. Huang, Degenerate scale problems and conservative
schemes of null-field method for Dirichlet problems of Laplace’s equation, manuscript, 2011.
[34] Z.C. Li, et al., Further investigation on the null-field method for Dirichlet problem of Laplace equa-
tion with very small circular hole, manuscript, 2011.
[35] Z.C. Li, et al., The Interior field method for Laplace’s equation on circular domains with circular
holes, manuscript, 2011.
[36] Z. C. Li, T. T. Lu, H. Y. Hu and A. H. D. Cheng, Trefftz and Collocation Methods, WIT
press, Southampton, Boston, January 2008.
[37] Z. C. Li, T. T. Lu, H. T. Huang and Cheng, A. H.-D., Trefftz, collocation and other boundary
methods, A comparison, Numer. Meth. for PDE, vol. 23, pp. 93–144, 2007.
[38] Z. C. Li, T. T. Lu, H. T. Hunag and A. H. -D. Cheng, (2009), Error analysis of Trefftz Method
for Laplace’s equations and its applications, CMES-Computer Medeling in Engineering & Sciences,
Vol. 52, pp. 39 - 81, 2009.
[39] Z. C. Li, H. J. Young, H. T. Huang, Y. P. Liu and A. H. -D. Cheng, Comparisons of fundamental so-
lutions and particular solutions for Trefftz methods, Engineering Analysis with Boundary Elements,
Vol. 34, pp. 248 – 258, 2010.
[40] C. B. Liem, T. L‥u and T. M. Shih, The Splitting Extrapolation Method, World Scientific,
Singapore, 1995.
[41] Z. C. Li, R. Mathon, and P. Sermer, Boundary methods for solving elliptic equations with singular-
ities, SIAM. J. Numer. Anal., Vol. 24, pp. 487-498, 1987.
[42] T. T. Lu, H. Y. Hu, and Z. C. Li, Highly accurate solutions of Motz’s and cracked-beam problems,
Engineering Analysis with Boundary Elements, Vol. 28, pp. 1378-1403, 2004.
[43] P. M. Morse and H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, Inc., New York,
1953.
[44] D. Palaniappan, Electrostatoics of two intersecting conducting cylinders, Math. Comput Modelling,
Vol. 36, pp. 821 – 830, 2002.
[45] J. E. Pasciak, Spectral and Pseudo spectral methods for advection equations, Math. Comp., Vol. 35,
pp. 1081–1092, 1980.
[46] D. H. Yu, Natural Boundary Integral Method and its Applications, Science Press/Kluwer
Academic Publishers, Beijing/New York, 2002.
[47] S. Z. Xie, Curved coordinate systems in Handbook of Modern Mathematical in Science
and Engineering in Chinese, Vol. I, Chapter 11, p. 433 -476, Zhong-Hua Institute Publishing,
China, 1985.