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

*(English)*Zbl 0883.65089Summary: In Part I [ibid. 143, No. 1-2, 113-154 (1997; reviewed above)], 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, 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,
\[
{\mathcal F}^{m,k}_{\varrho,h}u= \sum^{np}_{i=1} \langle u,{\mathcal K}_\varrho\rangle_i\check{\mathcal K}^h_\varrho(x- x_i,x)w_i.
\]
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. \({\mathcal C}^h_\varrho= 1\). 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 \(L_2\) error norm – they are synchronized.

##### MSC:

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

35J25 | Boundary value problems for second-order elliptic equations |

65N15 | Error bounds for boundary value problems involving PDEs |

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |

##### Keywords:

error bounds; moving least-square reproducing kernel method; Fourier analysis; Riesz bound; finite element method; multiresolution analysis; computational performance; MLSRK algorithm; convergence
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\textit{S. Li} and \textit{W. K. Liu}, Comput. Methods Appl. Mech. Eng. 139, No. 1--4, 159--193 (1996; Zbl 0883.65089)

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