Moving least-square reproducing kernel method Part II: Fourier analysis

In Part I of this work, the moving least-square reproducing kernel (MLSRK) method is formulated and implemented. Based on its generic construction, an m-consistency structure is discovered and the convergence theorems are established. In this part of the work, a systematic Fourier analysis is employed to evaluate and further establish the method. The preliminary Fourier analysis reveals that the MLSRK method is stable for sufficiently dense, non-degenerated particle distribution, in the sense that the kernel function family satisfies the Riesz bound. One of the novelties of the current approach is to treat the MLSRK method as a variant of the 'standard' finite element method and depart from there to make a connection with the multiresolution approximation. In the spirits of multiresolution analysis, we propose the following MLSRK transformation, (equation presented) The highlight of this paper is to embrace the MLSRK formulation with the notion of the controlled L_p-approximation. Based on its characterization, the Strang-Fix condition for example, a systematic procedure is proposed to design new window functions so they can enhance the computational performance of the MLSRK algorithm. The main effort here is to obtain a constant correction function in the interior region of a general domain, i.e. C^h_ρ = 1. This can create a leap in the approximation order of the MLSRK algorithm significantly, if a highly smooth window function is embedded within the kernel. One consequence of this development is the synchronized convergence phenomenon - a unique convergence mechanism for the MLSRK method, i.e. by properly tuning the dilation parameter, the convergence rate of higher-order error norms will approach the same order convergence rate of the L₂ error norm - they are synchronized.