A Meshless Galerkin Method Based On Boundary Integral Equations

The meshless (or meshfree) methods have drawn considerable attention in recent years. The main feature of this type of method is the absence of an explicit mesh, and the approximate solutions are constructed entirely based on a cluster of scattered nodes. Boundary integral equations (BIEs) are attractive computational techniques for linear and exterior problems as they can reduce the dimensionality of the original problem by one. The BIEs-based meshless method is an important branch of meshless methods.This dissertation first reviews the recent developments of the meshless methods by means of their discretization scheme, together with some comments on the advantages and also some existing problems to be further studied. The developments of mathematical theory of the meshless methods are introduced in detail, and also the basic theory is expatiated thoroughly. Then, based on the large amount of work by the pioneers, a new BIEs-based meshless Galerkin method----a Galerkin Boundary Node Method (GBNM) is proposed and applied successfully to solve problems in potential theory, linear elasticity and fluid mechanics.In the GBNM, an equivalent variational form of a BIE is used for representing the solution of boundary value problems, and the moving least-squares (MLS) approximation is employed to construct the trial and test functions of the variational form by a cluster of scattered boundary nodes instead of boundary elements. With the help of the MLS scheme, the GBNM is a boundary-type meshless method which requires only a nodal data structure on the bounding surface of the domain to be solved. A key advantage with the variational formulation is the GBNM can keep the symmetry and positive definiteness of the variational problems, a property that makes the method an ideal choice for coupling the finite element method or other established domain-type meshless methods such as the element-free Galerkin method for the problems with an unbounded domain. Besides, via multiplying the boundary function by a test function and integrating over the boundary, the boundary conditions of the original problem in the GBNM can be implemented exactly despite the MLS shape function lacking the delta function property.The current dissertation concentrates on the theoretical analysis and numerical applications of the GBNM. The main contents of this thesis are as follows.The first is the MLS approximation of a function on a generic boundary. The properties of the MLS approximation are studied comprehensively. Error estimates for the MLS approximation are deduced in Sobolev spaces when nodes and weight functions satisfy certain conditions. From the error analysis of the MLS scheme, we show that the error bound is directly related to the nodal spacing.The second is the GBNM for a general BIE which can be characterized as pseudo-differential operator equation. The BIE is firstly converted into a variational formulation, and then the MLS shape functions are used to generate the approximate space. A background cell structure is constructed for purposes of numerical integration. Based on the error estimates of the MLS approximation and the pseudo-differential operator theory, error estimates of the solution of the integral equation are established in Sobolev spaces. From the error analysis, it is shown that the error results mainly from the approximation of the boundary by the cell structure, on which the numerical integration to be carried out, and approximation of the boundary variable by the MLS scheme. When the BIE has some constraints, a Lagrangian multiplier can be introduced in the process of numerical approximation. The corresponding numerical implementation and error analysis are also provided.The third is the applications of the GBNM for the numerical solution of Laplace problems, biharmonic problems, linear elastic problems and Stokes problems. These boundary value problems are firstly reformulated as BIEs of the first kind, then the GBNM is used for obtaining the approximate solutions. The numerical implementations for these problems are described in depth. Total details of error analysis are given, and optimal asymptotic error estimates are obtained. When the integration cell structure is identical with the boundary, the estimates of the error in energy norms are also established. Numerical examples are presented to show the efficiency of the GBNM, and the numerical results are in consistency with the theoretical analysis.