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博碩士論文 etd-0526104-190036 詳細資訊
Title page for etd-0526104-190036
論文名稱
Title
橢圓型方程的Trefftz及Collocation法
The Trefftz and Collocation Methods for Elliptic Equations
系所名稱
Department
畢業學年期
Year, semester
語文別
Language
學位類別
Degree
頁數
Number of pages
189
研究生
Author
指導教授
Advisor
召集委員
Convenor
口試委員
Advisory Committee
口試日期
Date of Exam
2004-04-28
繳交日期
Date of Submission
2004-05-26
關鍵字
Keywords
奇異性、Trefftz法、結合法、collocation法、橢圓型方程
collocation method, elliptic equations, Trefftz method, singularities, combined method
統計
Statistics
本論文已被瀏覽 5793 次,被下載 2200
The thesis/dissertation has been browsed 5793 times, has been downloaded 2200 times.
中文摘要
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。
Abstract
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.
目次 Table of Contents
CHAPTER 0. Overview
CHAPTER 1. The CTM for Motz's and Cracked Beam Problems
CHAPTER 2. The CTM for Biharmonic Equations
CHAPTER 3. Collocation Methods
CHAPTER 4. Combinations of CM and FEM
CHAPTER 5. Radial Basis Collocation Methods
參考文獻 References
ibitem{AS70}
Abramowitz M. and Stegun I. A.,
{ f Handbook of Mathematical Functions with
Formulas, Graphs and Mathematical Tables},
Devor Publications, Inc, New York, 1970.

ibitem{AYB97}
Arad A., Yakhot A. and Ben-Dor G.,
{em A highly accurate numerical solution of
a biharmonic equation},
Numer. Methods for Partial Differential Equations,
Vol. 13, pp. 375-397, 1997.

ibitem{AW83}
Arnold D. N. and Wendland W. L.,
{em On the asymptotic convergence of collocation method},
Math. Comp., Vol. 41, No. 164, pp. 349-381, 1983.

ibitem{A89}
Atkison K. E.,
{ f An Introduction to Numerical Analysis},
John Wiley & Sons, 1989.

ibitem{B73}
Babuv{s}ka I.,
{ f The finite element method with Lagrangian multipliers}
Numer. Math., Vol. 20, pp. 179-192. 1973.

ibitem{BG02}
Babuv{s}ka I. and Guo B.,
{em Direct and inverse approximation theorems for the p-version of the
finite element method is the framework of weighted Besov spaces,
Part III: Inverse approximation theorems},
Technical Report, Department of Mathematics, University of Manitoba, 2002.

ibitem{BZ98}
Babuv{s}ka I. and Zhang Z.,
{em The partition of unity method for
the elastically supported beam},
Comput. Methods Appl. Mech. Engrg., Vol. 152, pp. 1-18, 1998.


ibitem{BDY90}
Bernardi C., Debit N. and Maday Y.,
{em Coupling finite element and spectral methods: First result},
Math. Comp., Vol. 54, pp. 21-39, 1990.

%10
ibitem{BM97}
Bernardi C. and Maday Y.,
{em Spectral methods in techniques of scientific computing (Part 2)},
{ f Handbook of Numerical Analysis}, Vol. V
(Edited by P. G. Ciarlet and J. L. Lions),
Elsevier Science, 1997.

ibitem{BL84}
Birkhoff G. and Lynch R. E.,
{ f Numerical Solution of Elliptic Equations},
SIAM, Philadelphia, 1984.

ibitem{B78}
Brebbia C. A.,
{ f The Boundary Element Methods for Engineering},
Prentch Press, London. 1978.

ibitem{BD89}
Brebbia C. A. and Dominguez J.,
{ f The Boundary Elements, In Introductory Course},
McGraw-Hill Book Company, New York, 1989.

ibitem{CHL85}
Canuto C., Hariharan S. I. and Lustman L.,
{em Spectral methods for exterior elliptic problems},
Numer Math. Vol. 46, pp. 505-520, 1985.

ibitem{CH87}
Canuto C., Hussaini M. Y., Quarteroni A. and Zang T. A.,
{ f Spectral Methods in Fluid Dynamics},
Springer-Verlag, New York, 1987.

ibitem{OC83}
Carey G. F. and Oden J. T.,
{ f Finite Elements, A Second Course, Vol, II},
Prentice Hall, Inc., Englewood Cliffs, N.J., 1983.

ibitem{LC98}
Chen C. H.,
{em Further study on Motz's problem}, Master thesis,
Department of Applied Mathematics,
National Sun Yat-sen University, Kaohsiung, 1998.


ibitem{C2002}
Cheng A. H.-D., Golberg M. A., Kansa E. J. and Zammito G.,
{em Exponential convergence and H-c multiquadrics collocation method for
partial differential equations},
Numer. Meth. for PDEs, Vol. 19, No. 5, pp. 571-594, 2003.

%20
ibitem{CJZ89}
Cheung Y. K., Jin W. G. and Zienkiewicz O. C.,
{em Direct solution procedure for solution of Harmonic
problems using complete, non-singular, Trefftz functions},
Communications in Applied Numerical Methods, Vol. 5,
pp. 159-169, 1989.

ibitem{CJZ91}
Cheung Y. K., Jin W. G. and Zienkiewicz O. C.,
{em Solution of Helmholtz equation by Trefftz method},
Inter. J. for Numerical Methods in Engineering, Vol. 32,
pp. 63-78, 1991.

ibitem{C89}
Chien W. Z.,
{ f Variational and Finite Element Methods} (in Chinese),
East Asia Publishing, Taipei, 1989.

ibitem{C91}
Ciarlet P. G.,
{em Basic error estimates for elliptic problems},
in Eds., P.G. Ciarlet and J. L. Lions,
{ f Finite Element Methods (Part I)}, pp. 17-352,
North-Holland, 1991.

ibitem{CH53}
Courant R. and Hilbert D.,
{ f Methods of Mathematical Physics},
Vol. I., Wiley Intersciences Publishers, New York, 1953.

ibitem{LM79}
Delves L. M.,
{em Global and regional methods},
in { f A Survey of Numerical Methods for Partial
Differential Equations}, Eds. by L. Gladwell & R. Wait,
Clarendon Press, Oxford, pp. 106-127, 1979.

ibitem{Dem97}
Demmel J. W.,
{ f Applied Numerical Linear Algebra},
SIAM, p. 132, 1997, Philadelphia.

ibitem{DG76}
Derrick W. R., and Grossman S. I.,
{ f Elementary Differential Equations with Applications},
Addison-Wesley Publishing Company, 1976.

ibitem{G76}
Do Groen P. P. N.,
{ f Singularly Perturbed Differential Operations of
Second Order}, Chap. III, Mathematisch Centum, Amsterdam, 1976

ibitem{FGW73}
Fix G. J., Gulati S. and Wakoff G. I.,
{em On the use of singular functions with finite element approximations},
J. Comp. Phys., Vol. 13, pp. 209-238, 1973.

ibitem{LF79}
Fox L.,
{em Finite differences and singularities on elliptic problem,}
in { f A Survey of Numerical Methods for Partial
Differential Equations}, Eds. by L. Gladwell & R. Wait,
Clarendon Press, Oxford, pp. 42-69, 1979.

%30
ibitem{F82}
Franke R.,
{em Scattered data interpolation tests of some methods},
Math. Comp. Vol. 38, pp. 181-200, 1982.

ibitem{FS98}
Franke R. and Schaback R.,
{em Solving partial differential equations by collocation using radial functions},
Applied Mathematics and Computation Vol. 93, pp. 73-82, 1998.

ibitem{GBP97}
Georgiou G. C., Boudouvis A. and Poullikkas A.,
Comparison of two methods for the computation of
singular solution in elliptic problem,
{em J. of Computational and Applied Mathematics,} Vol 79, pp. 277-289, 1997.


ibitem{GOS96}
Georgiou G. C., Olson L. G. and Smyrlis Y. S.,
{em A singular function boundary integral
method for the Laplace equation},
Communications in Numerical Methods in Engineering, Vol. 12,
pp. 127-134, 1996.

ibitem{G96}
Golberg M.,
{em Recent developments
in the numerical evaluation of partial solutions in the boundary element methods},
Applied Mathematics and Computation Vol. 75, pp. 91-101, 1996.

ibitem{GL89}
Golub G. H. and Loan C. F.,
{ f Matrix Computations (Sec Ed.)},
The Johns Hopkins University Press,
Baltimore and London, 1989.

ibitem{GO77}
Gottlieb D. and Orszag S. A.,
{ f Numerical Analysis of Spectral Methods: Theory and Applications},
SIAM, Philadelphia, 1977.

ibitem{GH81}
Gourgeon H. and Herrera I.,
{em Boundary methods c-complete systems for the biharmonic equations},
in { f Boundary Element Methods} (Irvine, Calif., 1981), pp. 431-441,
CML Publ., Springer, Berlin, 1981.

ibitem{GR80}
Gradshteyn S. and Ryzhik I. M.,
{ f Table of Integrals, Series, and Products,
Corrected and Enlarged Edition}, Academic Press,
New York, 1980.

ibitem{G85}
Grisvard P.,
{ f Elliptic Problems in Non-smooth Domains},
Pitman Advanced Publishing Program, Boston, 1985.

ibitem{G46}
Grunberg G.,
{em A new method of solution of certain boundary problems for
equations of mathematical physics
permitting of reparation of variables},
J. Phys., Vol. 10, pp. 301-320, 1946.

ibitem{H79}
Haidvogel D. B.,
{em The accurate solution of Poisson's equation by expansion
in Chebyshev polynomials},
J. Comput. Phys., Vol. 30, pp. 167-180, 1979.

%40
ibitem{H71}
Hardy R. L.,
{em Multiquadric equations of topography and other irregular surfaces},
J. of Geophysical Research, Vol. 76, pp. 1905-1915, 1971.

ibitem{Her84}
Herrera I.,
{ f Boundary Methods: An Algebraic Theory},
Pitman, 1984.

ibitem{H85}
Herrera I.,
{em Unified formulation of numerical methods. Part I.
Green's formulas for operators in discontinuous fields},
Numer. Methods for Partial Differential Equations,
Vol. 1, pp. 25-44, 1985.

ibitem{HD99}
Herrera I. and Diaz M.,
{em Indirect methods of collocation: Trefftz-Herrera collocation},
Numer. Methods for Partial Differential Equations,
Vol. 15, pp. 709-738, 1999.

ibitem{LH96}
Horng M. Y.,
{em Boundary approximation for Motz problem}, Master thesis,
Technical Report, Department of Applied Mathematics,
National Sun Yat-sen University, Kaohsiung, 1996.

ibitem{Hsu02}
Hsu C. H.,
{em Biharmonic boundary value problem with singularity},
Master thesis, Department of Applied Mathematics,
National Sun Yat-sen University, 2002.

ibitem{HL2002}
Hu H. Y. and Li Z. C.,
{em Combinations of
collocation methods for Poisson's equations},
Technical Report, Department of Applied Mathematics,
National Sun Yat-sen University, 2002.

ibitem{HL02}
Hu H. Y. and Li Z. C.,
{em Collocation methods for Poisson's equation},
Technical Report, Department of Applied Mathematics,
National Sun Yat-sen University, 2002.

ibitem{HLA02}
Hu H. Y., Li Z. C. and Cheng A. H.-D.,
{em Radial basis collocation method for elliptic boundary
value problem}, accepted for publication of Computer &
Mathematics with Applications,
March 2004.

%50
ibitem{HLW03}
Hu H. Y., Tsai H. S., Li Z. C. and S. Wang,
{em Particular solutions of singularly perturbed partial
differential equations with constant coefficients in
rectangular domains, Part II. computational aspects},
Technical Report, Department of Applied Mathematics,
National Sun Yat-sen University, 2004.

ibitem{JCZ93}
Jin W. G., Cheung Y. K. and Zienkiewicz O. C.,
{em Trefftz method for Kirchoff plate bending problems},
Inter. J. for Numerical Methods in Engineering, Vol. 36,
pp. 765-781, 1993.

ibitem{JC95}
Jin W. G. and Cheung Y. K.,
{em Trefftz direct method},
Advances in Engineering Software, Vol. 24 (1-3), pp. 65-69, 1995.

ibitem{JG86}
Jirousek J. and Guex L.,
{em The hybrid-Trefftz finite element model and
its application to plate bending},
Inter. J. for Numerical Methods in Engineering, Vol. 23,
pp. 651-693, 1986.

ibitem{KK95a}
Kamiya N. and Kita E.,
{em Trefftz method 70 years},
Advances in Engineering Software, Vol. 24 (1-3), pp. 1, 1995.

ibitem{K92a}
Kansa E. J. ,
{em Multiqudrics - A scattered data approximation
scheme with applications to computational fluid-dynamics - I
Surface approximations and partial derivatives},
Computer Math. Applic., Vol. 19, No.8&9, pp. 127-145, 1992.

ibitem{K92b}
Kansa E. J.,
{em Multiqudrics - A scattered data approximation
scheme with applications to computational fluid-dynamics - II
Solutions to parabolic, hyperbolic and elliptic partial differential equations},
Computer Math. Applic., Vol. 19, No.8&9, pp. 147-161, 1992.

ibitem{K92}
Karageorghis A.,
{em Modified methods of fundamental solutions
for harmonic and biharmonic problems with boundary singularities},
Numerical Methods for Partial Differential Equations, Vol. 8,
pp. 1-18, 1992.

ibitem{JK92}
Keiper J.,
{em Mathematica Numerics: Controlling the effects of
numerical errors in computation,}
Reprint from The Mathematica Conference, June, 1992, Boston, MA.

ibitem{KK95}
Kita E. and Kamiya N.,
{em Trefftz method, An overview},
Advances in Engineering Software, Vol. 24 (1-3), pp. 3-12, 1995.

%60
ibitem{Kol87}
Kolodziej J.,
{em Review of application of boundary
collocation methods in mechanics of continuous media},
SM Archives, Vol. 12 (4), pp. 187-231, 1987.

ibitem{Lef89}
Lefeber D.,
{ f Solving Problems with Singularities Using
Boundary Elements}, Computational Mechanics Publications,
Southampton, 1989.

ibitem{Lei98}
Leit~{a}o V. M. A.,
{em Application of multi-region Trefftz-collocation
to fracture mechanics},
Engineering Analysis with Boundary Elements,
Vol. 22, pp. 251-256, 1998.

ibitem{LS91}
Levitan B. M. and Sargsjan I. S.,
{ f Aturm-Liouville and Dirac Operators},
Kluwer Academic Publishers, 1991.

ibitem{Li97}
Li Z. C.,
{em Penalty combinations of Ritz-Galerkin and
finite difference method for singularity problems,}
J. Comput. Appl. Math., Vol. 81, pp. 1-17, 1997.

ibitem{Li98}
Li Z. C.,
{ f Combined Methods for Elliptic Equations with Singularities,
Interfaces and Infinities},
Kluwer Academic Publishers, Boston, London, 1998.

ibitem{LHHW03}
Li Z. C., Hu H. Y., Hsu C. H. and S. Wang,
{em Particular Solutions of singularly perturbed partial
differential equations with constant coefficients in
rectangular domains, Part I. convergence analysis},
Journal of Computational and Applied Mathematics,
Vol. 166, No. 1, pp.181-208, 2004.


ibitem{LTWM2004}
Li Z. C., Tsai H. S., Wang S. and Miller J. J. H.,
{em New models of singularly perturbed differential
equations with waterfalls solutions},
Technical Report, Department of Applied Mathematics,
National Sun Yat-sen University, 2004.


ibitem{LL2000}
Li Z. C. and Lu T. T.,
{em Singularities and treatments of
elliptic boundary value problems},
Mathematical and Computer Modeling, Vol. 31, pp. 79-145, 2000.

ibitem{LLH2004}
Li Z. C., Lu T. T. and Hu H. Y.,
{em Collocation Trefftz methods for biharmonic equations
with crack singularities},
Engineering Analysis with Boundary Elements, Vol. 28, No.1,
pp. 79-96, 2004.

ibitem{LM90a}
Li Z. C. and Mathon R.,
{em Boundary methods for elliptic problems on unbounded domains},
J. Comp. Phys., Vol. 89, pp. 414-431, 1990.

ibitem{LM90b}
Li Z. C. and Mathon R.,
{em Error and stability analysis of boundary methods for
elliptic problems with interfaces}, Math. Comp.,
Vol. 54, pp. 41-61, 1990.

%70
ibitem{LMS87}
Li Z. C., Mathon R. and Sermer P.,
{em Boundary methods for solving elliptic problem with singularities
and interfaces},
SIAM J. Numer. Anal., Vol. 24, pp. 487-498, 1987.

ibitem{LW2002}
Li Z. C. and Wang S.,
{em Penalty combinations of the Ritz-Galerkin method and
FEM for singularly perturbed differential equations},
Technical report, Department of Applied Mathematics,
National Sun Yat-sen University, 2002.

ibitem{LHL03}
Lu T. T., Hu H. Y. and Li Z. C.,
{em Highly accurate solutions of Motz's
and the cracked beam problems}, accepted for publication of
Engineering Analysis with Boundary Element, March 2004.


% ibitem{Lu01}
%Lu T. T. and Li Z. C.,
%{em The Cracked-beam problem by boundary approximation methods},
%Technical Report, Department of Applied Mathematics,
%National Sun Yat-sen University, 2002.

ibitem{LO93}
Lucas T. R. and Oh H. S.,
{em The method of auxiliary mapping for
the finite element solutions of elliptic problems containing singularities},
J. Comp. Phys. Vol. 1108, pp. 327 - 352, 1993.

ibitem{M92}
Madych W. R.,
{em Miscellaneous error bounds for
multiquadric and related interpolatory},
Computer Math. Applic., Vol. 24, No. 12, pp. 121-138, 1992.

ibitem{MN90}
Madych W. R. and Nelson S. A.,
{em Multivariate interpolation and
conditionally positive definite functions, I.}
Math. Comp. Vol. 54, pp. 211-230, 1990.

ibitem{MDTC01}
Mai-Duy N. and Tran-Cong T.,
{em Numerical solution of differential equations using
multiquadric radial basis function networks},
Neural Networks, Vol. 14, pp. 185-199, 2001.

ibitem{MDTC02}
Mai-Duy N. and Tran-Cong T.,
{em Mesh-free radial basis function network methods with domain
decomposition for approximation of functions and numerical solution
of Poisson's equations},
Engineering Analysis with Boundary Elements, Vol. 26, pp. 133-156, 2002.

ibitem{M86}
Marti J. T.,
{ f Introduction to Sobolev Spaces and Finite Element Solutions of
Elliptic Boundary Value Problems}, Academic Press, London, 1986.

%80
ibitem{M89}
Mercier B.,
{ f Numerical Analysis of Spectral Method},
Springer-Verlag, Berlin, 1989.

ibitem{MRS96}
Miller J. J. H., O'Roordan E. and Shiskin G. I.,
{ f Fitted Numerical Methods for Singular Perturbation Problems},
World Scientific, Singapore, 1996.

ibitem{MM47}
Motz M.,
{em The treatment of singularities of partial
differential equations by relaxation methods}, Quart Appl. Math,
Vol. 4, pp. 371-377, 1947.

ibitem{OR76}
Oden J. T. and Reddy J. N.,
{ f An Introduction to the Mathematical Theory of Finite Elements},
John Wiley & Sons, New York, 1976.

ibitem{OGS91}
Olson L. G.,Georgiou G. C. and Schultz W. W.,
{em An efficient finite element method
for treating singularities in Laplace's
equations}, J, Comp. Phys, Vol. 96, pp. 319-410, 1991.

ibitem{PW73}
Papamichael N. and Whiteman J. R.,
{em A numerical conformal transformation
method for harmonic mixed boundary value problems
in polygonal domain,}
J. Applied Math and Phys (ZAMP), Vol. 124, pp. 304-316, 1973.

ibitem{PK95}
Pathria D. and Karniadakis G. E.,
{em Spectral element methods for Elliptic problems in nonsmooth domains},
J. Comput. Phys. Vol. 122, pp. 83-95, 1995.

ibitem{PT90}
Piltner R. and Taylor R. L.,
{em The solution of plate blending problems with the aid of a
boundary element algorithm based on singular complex function},
in { f Boundary Elements XII}, pp. 437-445,
Proc. 12nd World Conf. on BEM, Sapporo, Japan, 1990.

ibitem{QV94}
Quarteroni A. and Valli A.,
{ f Numerical Approximation of Partial Differential Equation},
Springer-Verlag, Berlin, 1994.

ibitem{RST96}
Roos H. G., Stynes M., and Tobiska L.,
{ f Numerical Methods for Singularly Perturbed Differential Equations,
Convection-Diffusion and Flow Problems},
Springer, 1996.

%90
ibitem{RP75}
Rosser J. B. and Papamichael N.,
{em A power series solution of a harmonic mixed boundary value problem},
MRC, Technical report, University of Wisconsin, 1975.

ibitem{R87}
Russo R.,
{em A Green formulas for plane crack problems},
Boll. Un. Mat. Ttal. A(7), No. 1, pp. 59-67, 1987.

ibitem{Sc95}
Schaback R.,
{em Error estimates and condition numbers for radial basis function interpolation},
Advances in Computational Mathematics, Vol. 3, pp. 251-264, 1995.

ibitem{SFW79}
Schiff B. D., Fishelov D. and Whitman J. R.,
{em Determination of a stress intensity factor using local mesh refinement},
in { f The Mathematics of Finite Elements and Applications III},
pp. 55-64, Eds. by J. R. Whiteman, Academic Press, London, 1979.

ibitem{S94}
Shen J.,
{em Efficient Spectral-Galerkin method I. Direct solvers of second-
and fourth- order equations using Legendre polynomials},
SIAM. J. Sci. Comput., Vol. 14, pp. 1489-1505, 1994.

ibitem{S95}
Shen J.,
{em Efficient Spectral-Galerkin method II. Direct solvers of second-
and fourth- order equations using Chebyshev polynomials},
SIAM. J. Sci. Comput., Vol. 15, pp. 74-87, 1995.

ibitem{S97}
Shen J.,
{em Efficient Spectral-Galerkin method III.
polar and cylindrical geometries},
SIAM. J. Sci. Comput., Vol. 18, pp. 1583-1604, 1997.

ibitem{S96}
Sneddon G. E.,
{em Second-order spectral differentiation matrices},
SIAM J. Numer. Anal. Vol. 33, pp. 2468-2487, 1996.

ibitem{S63}
Sobolev S. L.,
{ f Application of Fundamental Analysis in Mathematical Physics},
AMS, Providence, RI, 1963.

ibitem{SF73}
Strang G. and Fix G. J.,
{ f An Analysis of the Finite Element Method},
Prentice-Hall, 1973.

ibitem{S93}
Strenger F.,
{ f Numerical Methods based on Sinc and Analytic Functions},
Springer-Verlag, Berlin, 1993.

%100
ibitem{Lu01a}
Tang L. D.,
{em The Cracked-beam and Related Singularity Problems},
Master Thesis, Department of Applied Mathematics,
National Sun Yat-sen University, 2001.

ibitem{RW76}
Thatcher R. W.,
{em The use of infinite grid refinement at
singularities in the solution of Laplace's equation,} Numer. Math,
Vol. 25, pp. 163-178, 1976.

ibitem{T26}
Trefftz E.,
{em Ein Gegenstuck zum Ritz'schen Verfahren},
proc, 2nd Int. Cong. Appl. Mech., Zurch, pp. 131-137, 1926.

ibitem{V94}
Volkov E. A.,
{ f Block Method for Solving the Laplace Equation and
for Constructing Conformal Mappings},
CRC Press, Boca Raton, London, 1994.

ibitem{W2002}
Wendland H.,
{em Meshless Galerkin methods using radial basis functions},
Math. Comp., Vol. 68, No. 228, pp. 1521-1531, 1999.

ibitem{W78}
Whitman J. R.,
{em Numerical treatment of a problem from linear fracture mechanics},
in { f Numerical Methods in Fracture Mechanics},
pp. 128-136, Eds. by D. R. J. Owen and A. R. Luxmoore,
University of Wales, Swansea, 1978.

ibitem{WP72}
Whiteman J. R. and Papamichael N.,
{em Treatment of harmonic
mixed boundary by conformal transformation method,} ZAMP, Vol. 23,
pp. 655-664, 1972.

ibitem{W68}
Wigley N. M.,
{em On a method to subtract off a singularity at a corner for the
Dirichlet or Neumann problem},
Math. Comp., Vol. 23, pp. 395-401, 1968.

ibitem{NM88}
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