Moving least-square reproducing kernel methods (I) Methodology and convergence
Moving least-square reproducing kernel methods (I) Methodology and convergence
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DOI：
10.1016/s0045-7825(96)01132-2
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发表时间：
1997-04
影响因子：
7.2
通讯作者：
Wing Kam Liu;Shaofan Li;T. Belytschko
中科院分区：
文献类型：
--
作者：
Wing Kam Liu;Shaofan Li;T. Belytschko
在移动最小二乘再生核(MLSRK)表示的框架下，提出了移动最小二乘插值法。在这项研究中，移动最小二乘插值函数的构造过程被使用再生核公式的概念简化，作为早期离散方法的推广，它为单位分解建立了连续的基础。这种新的公式具有简单、易于实施的特点。此外，所提出的再生核公式不仅能够在不规则的粒子分布上精确地再现任意m次多项式，而且可以作为一种投影算子，在全局上以最佳精度逼近任意光滑函数。在这一贡献中，找到了一个一般的m相容关系，这是MLSRK近似的本质性质。给出了一个内插误差估计来评估逼近的收敛速度。证明了对于足够光滑的函数，其关于采样值的插值级数展开式将在Soblev范数下收敛到原函数。作为一种无网格法，收敛速度是通过一个新的控制变量--窗函数的伸缩参数ρ来测量的，而不是有限元分析中通常采用的网格尺寸h。为了说明这一过程，我们用Galerkin方法证明了二阶椭圆型微分方程数值解的收敛。在数值算例中，用该方法求解了一个两点边值问题，得到了关于不同范数的最优收敛速度。
This paper formulates the moving least-square interpolation scheme in a framework of the so-called moving least-square reproducing kernel (MLSRK) representation. In this study, the procedure of constructing moving least square interpolation function is facilitated by using the notion of reproducing kernel formulation, which, as a generalization of the early discrete approach, establishes a continuous basis for a partition of unity. This new formulation possesses the quality of simplicity, and it is easy to implement. Moreover, the reproducing kernel formula proposed is not only able to reproduce any mth order polynomial exactly on an irregular particle distribution, but also serves as a projection operator that can approximate any smooth function globally with an optimal accuracy. In this contribution, a generic m-consistency relation has been found, which is the essential property of the MLSRK approximation. An interpolation error estimate is given to assess the convergence rate of the approximation. It is shown that for sufficiently smooth function the interpolant expansion in terms of sampled values will converge to the original function in the Sobolev norms. As a meshless method, the convergence rate is measured by a new control variable—dilation parameter ρ of the window function, instead of the mesh size h as usually done in the finite element analysis. To illustrate the procedure, convergence has been shown for the numerical solution of the second-order elliptic differential equations in a Galerkin procedure invoked with this interpolant. In the numerical example, a two point boundary problem is solved by using the method, and an optimal convergence rate is observed with respect to various norms.