Researches On The Convergence And Error Analysis Of Meshless Methods

The meshless (or meshfree) method has been a hot direction and the development trend of numerical methods for science and engineering problems in recent years. As a new kind of numerical method, there was a little investigation on corresponding mathematical theory for meshless method. Without the corresponding of mathematical theory, the research and application of meshless method would be certainly restricted. The finite element method has got great development for its solid foundation of mathematical theory. Although some theories of finite element method can be adapted to meshless method, it also needs to develope the theory which is suitable for meshless method itself, such as convergence, stability, error estimates, and etc.In this dissertation, the convergence and error estimate of some meshless methods have been researched and analyzed. The main contents of this thesis are as follows.The moving least-square approximation is a method which is used to obtain the trial function of element-free Galerkin method. Error estimates for the moving least-square approximation in Sobolev spaces are obtained in multidimensional case. We present the optimal order error estimates for approximation function and the corresponding high order derivatives when nodes and shape functions satisfy some conditions. From the error analysis of this method, we show that the error bound of the numerical solution is directly related to the radii of the weight functions.The element-free Galerkin method is one of the meshless methods, which is most widely researched and applied. Error estimates of element-free Galerkin method for potential problem are analyzed. We put forward that error estimates have optimal order when nodes and shape functions satisfy some conditions. From the error analysis, it is shown that the error bound of the potential problem is directly related to the radii of the weight functions.Based on the error estimates of moving least-square approximation, error estimates of element-free Galerkin method for elasticity are analyzed. And the relationship between error estimates and radii of weight functions is also presented.The parabolic partial differential equation is an important type of mathematical physical equations. It can describe the problems of heat conduction, seepage flow and diffusion, et al. Up to now, finite difference method, finite element method and boundary element method are common numerical methods to obtain numerical solutions of parabolic partial differential equation. In this paper, the element-free Galerkin method for heat conduction problem and the corresponding error estimates are proposed. The semi-discrete and full-discrete element-free Galerkin approximation for linear and nonlinear heat conduction problems and corresponding error estimates are discussed. From the error analysis, it is shown that the error of semi-discrete element-free Galerkin approximation is related to the radii of weight functions, and the error of full-discrete approximation is not only related to the radii of weight functions, but also related to the step length of time.Based on the error estimates of moving least-square approximation, the error estimates for the finite point method are obtained. From the error analysis of the finite point method, we show that the error bound of the numerical solution is directly related to the radii of the weight functions and the condition number of the coefficient matrix.Inverse problem is difficult to be solved in scientific research. It is widely applied in the aerospace, nuclear physics, metallurgy and other fields. In this paper, the finite point method is used to obtain numerical solutions of one-dimensional and two-dimensional inverse heat conduction problems with a source parameter, and the corresponding discrete equations are obtained. Comparing with the numerical methods based on mesh, the finite point method only needs the scattered nodes instead of meshing the domain of the problem. The finite point method is a meshless method in which the moving least-square approximation is used to form the meshless approximation functions, and the collocation method is used to discretize the governing partial differential equations. The finite point method has some advantages of simple numerical procedures, low computation cost and arbitrary nodes. In order to show the validity of results given in the dissertation, Corresponding MATLAB codes have been written. Some numerical examples are provided to show the correct of the above theorems and results.The paper presents the error analysis theory of the element-free Galerkin method and the finite point method. The researches of this thesis provide some matheatic theories for the further development and application of meshless methods.