在這篇論文裡，分析基本解法推廣到重(雙)調和方程。推導出傳統基解法和Almansi的基本解法誤差界在單連通域裡。在高度平滑和有限可微的解分別可得到指數和多項式的收斂速度。並且推導條件數界在圓盤區域，證明為指數的增長率。這篇論文首次提供了嚴謹的CTM解重調和方程的分析，而準確性和穩定性的本質是類似於拉普拉斯方程。
我們實行了光滑和奇異問題的數值實驗。數值計算的結果與理論分析相吻合。當可以找到滿足雙調和方程的特解，根據數值結果特解法總是優於基本解法。但是，如果奇異點附近的特解不能被發現的話，適當地在附近加密配置點和greedy adaptive技術可以使用。似乎greedy adaptive技術可為奇異問題提供一個更好的解決辦法。此外，Almansi基本解法的數值解不論在精準度和穩定性都比傳統基本解法略好。由於分析可簡單直接地從調和方程的基本。

In this thesis, the analysis of the method of fundamental solution(MFS) is expanded for biharmonic equations. The bounds of errors are derived for the traditional and the Almansi's approaches in bounded simply-connected domains. The exponential and the polynomial convergence rates are obtained from highly and finite smooth solutions, respectively. Also the bounds of condition number are derived for the disk domains, to show the exponential growth rates. The analysis in this thesis is the first time to provide the rigor analysis of the CTM for biharmonic equations, and the intrinsic nature of accuracy and stability is similar to that of Laplace's equation.
Numerical experiment are carried out for both smooth and singularity problems. The numerical results coincide with the theoretical analysis made. When the particular solutions satisfying the biharmonic equation can be found, the method of particular solutions(MPS) is always superior to MFS, supported by numerical examples. However, if such singular particular solutions near the singular points can not be found, the local refinement of collocation nodes and the greedy adaptive techniques can be used. It seems that the greedy adaptive techniques may provide a better solution for singularity problems. Beside, the numerical solutions by Almansi's approaches are slightly better in accuracy and stability than those by the traditional FS. Hence, the MFS with Almansi's approaches is recommended, due to the simple analysis, which can be obtained directly from the analysis of MFS for Laplace's equation.

1 Introduction . . . . . . . . . . . . 1
2 Error Analysis . . . . . . . . . . . . 1
2.1 Description of MFS . . . . . . . . . . . . 1
2.2 Analysis for the Almansi’s approaches . . . . 3
3 Preliminary Lemmas . . . . . . . . . . . . 6
4 Main Theorems . . . . . . . . . . . . 17
5 Stability Analysis of the MFS for Bihormonic Equation on Circular Domains. . . . . . . . . . . . 23
5.1 Approaches for eigenvalues . . . . . . . . . . 23
5.2 Eigenvalue λk(Φ) . . . . . . . . . . . . . . . 30
5.3 Eigenvalues λk(DΦ) . . . . . . . . . . . . . . 33
5.4 Eigenvalue of MFS . . . . . . . . . . . . . . . . 37
6 Numerical Experiments . . . . . . . . . . . . 44
6.1 Plate Bending Problem with Smooth Solutions . . . . . . . . . . . 44
6.2 Plate Bending Problem with Crack Singularity . . . . . . . . . . . . . . 49
6.2.1 MFS by local refinements of collocation nodes . . . . . . . . . . . 49
6.2.2 Greedy Adaptive Techniques . . . . . . . 54
7 Concluding Remarks . . . . . . . . . . . . 60

[1] Bogomolny, A., Fundamental solutions method for elliptic boundary value problems,
SIAM, J. Numer, Anali vol. 22, pp.644-669, 1985.
[2] Chen, J.T., Wu, C.S, Lee, Y.T. and Chen, K.H., On equivalence of Trefftz method
of fundamental solutions for Laplace and biharmonic equations, Computer & Mathematics
with Applications, vol. 53, pp. 851-879, 2007.
[3] Davis, P Circulant Matrices, John Wiley & Sons, New York, 1979.
[4] Fichera, Linear elliptic equations of higher order in two independent variables and
singular integral equations with applications to an isotropic in homogeneous elasticity
in Partial Differential Equations ad Continuum Mechanices, ed. R.E., Langer
University of Wisconsin Press, Madison, pp. 55-80, 1961.
[5] Fairweather, G. and Karageorghis, A., The method of fundamental solutions for
elliptic boundary value problems, Advances in Computational Mathematics, vol. 9,
pp. 69-95, 1998.
[6] Karageorghis, A. and Fairweather, G., The Almansi method of fundamental solutions
for solving biharmonic problems, Iuter, J. Numer. Meth in Engrg, vol.26, pp.
1665-1682, 1988.
[7] Karageorghis, A. and Fairweather, G., The simple layer potential method of fundamental
solutions for certain biharmonic problems, Iuter, J. Methods Fluids, vol. 9,
pp. 1221-1234, 1989.
[8] Karageorghis, A. and Fairweather, G., The method of fundamental solutions for the
numerical solution of biharmonic equation, J. Comp. Physics, vol.69, pp. 434-459,
1998.
[9] Li, Z. C., Huang, S. L., J. Hunag and Huang, H. T., Stability analysis of method of
fundamental solutions for mixed boundary problems of Laplace’s equation, Technical
report, Department of Applied Mathematics, National Sun Yet-sen University,
Kaohsiung, 2006.
[10] Li, Z.C., Method of fundamental solutions for annular shaped domains, Technical
report, Department of Applied Mathematics, National Sun Yat-sen University,
Kaohsiung, Taiwan. 2007.
[11] Li, Z.C., An analysis of method of fundamental solution for neumann problems of
Laplace’s equation, Technical report, Department of Applied Mathematics, National
Sun Yat-sen University, Kaohsiung, Taiwan. 2007.
[12] Li, Z.C., Chien, C.S. and Huang, H.T., Effective condition number for finite differece
method. J. Comput. Appl. Math., vol.198 pp.208-235, 2007.
[13] Li, Z.C., Huang, S.L, Huang H.T, and Huang J., Stability analysis of method of
fundamental solutions for mixed boundary value problems of Laplace’s equations,
Technical report, Department of Applied Mathematics, National Sun Yat-sen University,
Kaohsiung, Taiwan. 2007.
[14] Li, Z.C., Lu, T.T. and Cheng, A.H.-D., Error analysis of Trefftz methods for
Laplace’s equations with singularity problems, Technical report, Department of Applied
Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan. 2007.
[15] Li, Z.C, Lu, T.T and Hu H.Y, The collocation Trefftz method for biharmonic equations
with singularities, Engineering Analysis with Boundary Elements, Vol.28, pp.
79-96, 2004.
[16] Li, Z.C, Lu, T.T, Hu H.Y. and Cheng, A.H.-D., Trefftz and Collocation Meth-
ods, WIT, Southsampton, Boston, 2008.
[17] Ling, L., Opfer, R. and Schaback, R., Results on meshless collocation techniques,
Engineering Analysis with Boundary Elements, vol.30, pp.247-253, 2006.
[18] Li, Z.C, Lu, T.T and Yimin Wei, Effective Condition Number of Trefftz Methods for
Biharmonic Equations with Crack Singularities , Technical report, Department of
Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan. 2008.
[19] Schaback, R, Adaptive numerical solution of MFS systems, a plenary talk at the first
Inter. Workshop on the Method of Fundamental Solutions (MFS2007), Ayia Napa,
Cyprus, June 11-13, 2007.
[20] Watson, G.N., A Treatise on the Theory of Bessel function, Cambrige at the University Press, 1962.