| At present, the meshless method is popular in the computing science with its unique advantages. It bases on the numerical solution of the nodes, which will be enough to solve partial differential equations, and overcomes the dependence on the finite element method as the grid. Compared with the finite element method, the meshless method is a new kind of numerical method, which requires a strong mathematical theory basis, such as convergence, stability, error estimate, and so on. Therefore, studying on the theory of the meshless method has its far-reaching significance.Moving least-squares(MLS) method, a kind of numerical method which generate the kind of shape function in the meshless method, is widely being applied. In this dissertation, the theory of error and application of the MLS has been researched and analyzed. The main contents of this thesis are as follows.Error estimates for MLS approximation in Sobolev spaces are obtained in two dimensional cases. The author studied the weak smooth function and strong smooth function, respectively, then analyzed the error estimates for MLS approximation in Sobolev spaces in norm. Thus, the analysis of the results is more universal, and has important theoretic value.The application of MLS has been reached on the analysis of its error estimates. In the first place, the dissertation provides the comparative study on curved surface fitting of two kinds of MLS. In the next place, based on the MLS method, the error estimates of the Interpolating Element-Free Galerkin(IEFG) method for potential problem are studied on the analysis of its error estimates. In the meantime, the dissertation provides the Interpolating Element-Free Galerkin(IEFG) method as the numerical examples for the error estimates. In order to prove the validity of results given in the dissertation, corresponding MATLAB codes have been written. Sevarial numerical examples are given to demonstrate the correctness of this theory. |
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