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.

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.

In this thesis, the boundary errors are defined for the NFM to explore the convergence rates, and the condition numbers are derived for simple cases to explore numerical stability. The optimal convergence (or exponential) rates are discovered numerically. This thesis is also devoted to seek better choice of locations for the field nodes of the FS expansions. It is found that the location of field nodes Q does not affect much on convergence rates, but do have influence on stability. Let £_ denote the distance of Q to ∂S. The larger £_ is chosen, the worse the instability of the NFM occurs. As a result, £_ = 0 (i.e., Q ∈ ∂S) is the best for stability. However, when £_ > 0, the errors are slightly smaller. Therefore, small £_ is a favorable choice for both high accuracy and good stability. This new discovery enhances the proper application of the NFM. However, even for the Dirichlet problem of Laplace¡¦s equation, when the logarithmic capacity (transfinite diameter) C_£F = 1, the solutions may not exist, or not unique if existing, to cause a singularity of the discrete algebraic equations. The problem with C_£F = 1 in the BEM is called the degenerate scale problems. The original explicit algebraic equations do not satisfy the conservative law, and may fall into the degenerate scale problem discussed in Chen et al. [15, 14, 16], Christiansen [35] and Tomlinson [42]. An analysis is explored in this thesis for the degenerate scale problem of the NFM. In this thesis, the new conservative schemes are derived, where an equation between two unknown variables must satisfy, so that one of them is removed from the unknowns, to yield the conservative schemes. The conservative schemes always bypasses the degenerate scale problem; but it causes a severe instability. To restore the good stability, the overdetermined system and truncated singular value decomposition (TSVD) are proposed. Moreover, the overdetermined system is more advantageous due to simpler algorithms and the slightly better performance in error and stability. More importantly, such numerical techniques can also be used, to deal with the degenerate scale problems of the original NFM in [15, 14, 16]. For the boundary integral equation (BIE) of the first kind, the trigonometric functions are used in Arnold [3], and error analysis is made for infinite smooth solutions, to derive the exponential convergence rates. In Cheng¡¦s Ph. Dissertation [18], for BIE of the first kind the source nodes are located outside of the solution domain, the linear combination of fundamental solutions are used, error analysis is made only for circular domains. So far it seems to exist no error analysis for the new NFM of Chen, which is one of the goal of this thesis. First, the solution of the NFM is equivalent to that of the Galerkin method involving the trapezoidal rule, and the renovated analysis can be found from the finite element theory. In this thesis, the error boundary are derived for the Dirichlet, the Neumann problems and its mixed types. For certain regularity of the solutions, the optimal convergence rates are derived under certain circumstances. Numerical experiments are carried out, to support the error made.