The generalized finite difference method (GFDM) is a typical meshless collocation method based on the Taylor series expansion and the moving least squares technique. In this paper, we first provide theoretical results of the meshless function approximation in the GFDM. Properties, stability and error estimation of the approximation are studied theoretically, and a stabilized approximation is proposed by revising the computational formulas of the original approximation. Then, we provide theoretical results consisting of error bound and condition number of the GFDM. Numerical results are finally provided to confirm these theoretical results.

中文翻译：

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广义有限差分法的理论分析

广义有限差分法（GFDM）是一种典型的基于泰勒级数展开和移动最小二乘技术的无网格配置方法。在本文中，我们首先提供了 GFDM 中无网格函数逼近的理论结果。对近似的性质、稳定性和误差估计进行了理论研究，通过修改原近似的计算公式，提出了一种稳定的近似。然后，我们提供了由 GFDM 的误差界限和条件数组成的理论结果。最后提供了数值结果来证实这些理论结果。