The localized method of fundamental solutions (LMFS) is an efficient meshless collocation method that combines the concept of localization and the method of fundamental solutions (MFS). The resultant system of linear algebraic equations in the LMFS is sparse and banded and thus, drastically reduces the storage and computational burden of the MFS. In the LMFS, the moving least square (MLS) approximation, based on fundamental solutions, is used to construct approximate solution at each node. In this paper, this fundamental solutions-based MLS approximation, named as an augmented MLS (AMLS) approximation, is generalized to any point in the computational domain. Computational formulas, theoretical properties and error estimates of the AMLS approximation are derived. Then, taking Laplace equation as an example, this paper sets up a framework for the theoretical error analysis of the LMFS. Finally, numerical results are presented to verify the efficiency and theoretical results of the AMLS approximation and the LMFS. Convergence and comparison researches are conducted to validate the accuracy, convergence and efficiency.

基本解的局部化方法（LMFS）是一种有效的无网格配置方法，结合了本地化的概念和基本解的方法（MFS）。LMFS中所得的线性代数方程组稀疏且有条带，因此大大减少了MFS的存储和计算负担。在LMFS中，基于基本解的移动最小二乘（MLS）近似用于构造每个节点的近似解。在本文中，基于基本解决方案的MLS近似（称为增强MLS（AMLS）近似）被推广到计算域中的任何点。推导了AMLS近似的计算公式，理论性质和误差估计。然后，以拉普拉斯方程为例，本文为LMFS的理论误差分析建立了框架。最后，给出了数值结果，以验证AMLS近似和LMFS的效率和理论结果。进行了收敛和比较研究以验证准确性，收敛性和效率。