Most of reports deal with bounded simply-connected domains; only a few involve in exterior and annular problems (Chen et al. [3], Katsuroda[10] and Ushijima and Chibu [30]). For exterior problems of Laplace's equations, there exist two kinds of infinity conditions, (1) |u|≤C and (2) u=O( ln r), which must be complied with by the fundamental solutions chosen. For u=O(ln r), the traditional fundamental solutions can be used. However, for |u|≤C, new fundamental solutions are explored, with a brief error analysis. Numerical experiments are carried out to verify the theoretical analysis made. Numerical experiments are also provided for annular domains, to show that the method of fundamental solutions (MFS) is inferior to the method of particular solutions (MPS), in both accuracy and stability. MFS and MPS are classified into the Trefftz method (TM) using fundamental solutions (FS) and particular solutions (PS), respectively. The remarkable advantage of MFS over MPS is the uniform $ln|overline{PQ_i}|$, to lead to simple algorithms and programming, thus to save a great deal of human power. Hence, we may reach the engineering requirements by much less efforts and a little payment. Besides, the crack singularity in unbounded domain is also studied. A combination of both PS and FS is also employed, called combination of MFS. The numerical results of MPS and combination of MFS are coincident with each other. The study in this thesis may greatly extend the application of MFS from bounded simply-connected domains to other more complicated domains.

In the thesis, the error and stability analysis is made for the 2D Helmholtz equation by the method of fundamental solutions (MFS) using both Bessel and Neumann functions. The bounds of errors in bounded simply-connected domains are derived, while the bounds of condition number are derived only for disk domains. The MFS using Bessel functions is more efficient than the MFS using Neumann functions. Interestingly, for the MFS using Bessel functions, the radius R of the source points is not necessarily larger than the maximal radius r_max of the solution domain. This is against the traditional condition: r_max < R for MFS. Numerical experiments are carried out to support the analysis and conclusions made.

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.

The singular solutions for linear elastostatics at corners are essential in both theory and computation. In this thesis, we seek new singular solutions for corners with the fixed (displacement), the free stress (traction) boundary conditions, and their mixed types, and to explore their corner singularity and provide the algorithms and error estimates in detail. The singular solutions of linear elastostatics are derived, and a number of new models of corner and crack singularity are proposed. Effective numerical methods, such as the collocation Trefftz methods (CTM), the method of fundamental solutions (MFS), the method of particular solutions (MPS) and their combinations: the so called combined method, are developed. Such solutions are useful to examine other numerical methods for singularity problems in linear elastostatics. This thesis consists of three parts, Part I: Basic approaches, Part II: Advanced topics, and Part III: Mixed types of displacement and traction conditions. Contents of Parts I and II have been published in [47,82]. In Part I, the collocation Trefftz methods are used to obtain highly accurate solutions, where the leading coefficient has 14 (or 13) significant digits by the computation with double precision. In part II, two more new models (symmetric and anti-symmetric) of interior crack singularities are proposed, for the corner and crack singularity problems, the combined methods by using many fundamental solutions, but by adding a few singular solutions are proposed. Such a kind of combined methods is significant for linear elastostatics with corners (i.e., the L-shaped domain), because the singular solutions can only be obtained by seeking the power £hk of r£hk numerically. Hence, only a few singular solutions used may greatly simplify the numerical algorithms; Part III is a continued study of Parts I and II, to explore mixed type of displacement and free traction boundary conditions. To our best knowledge, this is the first time to provide the particular solutions near the corner with mixed types of boundary conditions and to report their numerical computation with different boundary conditions on the same corner edge in linear elastostatics. This thesis explores corner singularity and its numerical methods, to form a systematic study of basic theory and advanced computation for linear elastostatics.