| This dissertation is a technical document on a numerical algorithm-Moving Least Square Reproducing Kernel Method, a prototype of meshless methods. The thesis is divided into three parts. Part I furnishes a complete treatise on the methodology, interpolation estimate, and convergence of the method. The main contributions of this part are: (1) The derivation of the canonical form of moving least square approximation; (2) The discovery of the differential consistency condition for the reproducing kernel method; (3) The establishment of interpolation error estimate.; In Part II, a systematic Fourier analysis is conducted to evaluate the method in the sense of controlled {dollar}Lsp2{dollar}-approximation. In consequence, a genuine reproducing kernel approximation is derived and the notion of synchronized convergence is proposed in the first time. In the context of reproducing kernel method, the phenomenon of the synchronized convergence is the consequences of the generalized moment theorem, truncation error representation, and the Strang-Fix condition. The moral of the story is simple: it is the smoothness of the window function embedded that contributes the extra accuracy for the interpolation scheme. In turn, it offers a partial explanation why the reproducing kernel method has better accuracy than its counterpart--finite element method with the same polynomial basis.; In Part III, a novel synchronized reproducing kernel interpolant is proposed, which is constructed by a cluster of wavelets--a wavelet packet. The wavelet packet, including the scaling function kernel and the higher scale wavelets, is generated in a moving least square reproducing procedure. These wavelets are a new class of wavelet functions, which are different from the available wavelets in the literature. It is the first of its kind. The same construction procedure could be extended to other interpolation procedures. As a first step application, a so-called wavelet Petrov-Galerkin method is proposed in the first time and used in computation to stabilize unstable numerical computation for some pathological problems, such as advective-diffusive problem and the Stokes flow problem. For advective-diffusive equation, the stability and the convergence of Petrov-Galerkin method is given. |
