The Lagrange multiplier used for the Dirichlet condition is well known in mathematics community, and the Lagrange multiplier used for the Neumann condition is popular for the Trefftz method in engineering community, in particular for elasticity problems. The latter is called the Hybrid Trefftz method (HTM). However, it seems to export no analysis for HTM. This paper is devoted to error analysis of the HTM for −£Gu + cu = 0 with c = 1 or c = 0. Error bounds are derived to provide the optimal convergence rates. Numerical experiments and comparisons between two kinds of Lagrange multipliers are also reported. The analysis in this paper can also be extended to the HTM for elasticity problems.

The Lagrange multiplier used for the displacement (i.e., Dirichlet) condition is well known in mathematics community (see [1, 2, 10, 18]), and the Lagrange multiplier used for the traction (i.e., Neumann)condition is popular for the Trefftz method for elasticity problems in engineering community, which is called the Hybrid Trefftz method (HTM). However, it seems to export no analysis for HTM. This paper is devoted to error analysis of the HTM for elasticity problems. Numerical experiments are reported to support the analysis made.

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.

none

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 literature Trefftz method normally has geometric (exponential) convergence. Recently many scholars have found that spectral method in some cases can converge faster than exponential, which is called super-geometric convergence. Since Trefftz method can be regarded as a kind of spectral method, we expect it might possess super-geometric convergence too. In this thesis, we classify all types of super-geometric convergence and compare their speeds. We develop a method to decide the convergent type of given error data. Finally we can observe in many numerical experiments the super-geometric convergence of Trefftz method to solve Helmholtz boundary value problems.

For Laplace's eigenvalue problems, this thesis presents new algorithms of the Trefftz method (i.e. the boundary approximation method), which solve the Helmholtz equation and then use a iteration process to yield approximate eigenvalues and eigenfunctions. The new iteration method has superlinear convergence rates and gives a better performance in numerical testing, compared with the other popular methods of rootfinding. Moreover, piecewise particular solutions are used for a basic model of eigenvalue problems on the unit square with the Dirichlet condition. Numerical experiments are also conducted for the eigenvalue problems with singularities. Our new algorithms using piecewise particular solutions are well suited to seek very accurate solutions of eigenvalue problems, in particular those with multiple singularities, interfaces and those on unbounded domains. Using piecewise particular solutions has also the advantage to solve complicated problems because uniform particular solutions may not always exist for the entire solution domain.

Boundary approximation method, also known as the collocation Trefftz method in engineering, is used to solve Laplace boundary value problem on rectanglular domain. Suppose the particular solutions are chosen for the whole domain. If there is no singularity on other vertices, it should have exponential convergence. Otherwise, it will degenerate to polynomial convergence. In the latter case, the order of convergence has some relation with the intensity of singularity. So, it is easy to design models with desired convergent orders. On a sectorial domain, when one side of the boundary conditions is a transcendental function, it needs to be approximated by power series. The truncation of this power series will generate an artificial singularity when solving Laplace equation on polygon. So it will greatly slow down the expected order of convergence. This thesis study how the truncation error affects the convergent speed. Moreover, we focus on the transition behavior of the convergence from one order to another. In the end, we also apply our results to boundary approximation method with enriched basis.

The error analysis is made for the simple annular domain with the circular boundaries having the same origin. The error bounds are derived, and the optimal convergence rates can be archived. For circular domains with circular boundaries, the same convergence rates can be achieved by strict proof. Moreover, the NFM algorithms and its conservative schemes can be applied to very small holes, which are difficult for other numerical methods to handle. Both the NFM and the collocation Trefftz method(CTM) are used for very small circular holes, the CTM is superior to the NFM in accuracy and stability.

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.