中文摘要：

配点类无网格法需要计算近似函数的二阶导数，因而在移动最小二乘（MLS）近似中至少要采用二次基函数。本文利用Voronoi图对双重点移动最小二乘近似法进行了改进，建立了基于Voronoi图的双重点移动最小二乘近似（VDG），并利用加权最小二乘法离散微分方程，导出了双重点最小二乘配点无网格法（MDGLS）。该方法将求解域用节点离散，并以节点为生成点建立Voronoi图，取Voronoi多边形的顶点为辅助点。近似函数及其二阶导数的计算过程可分解为两个步骤：首先用场函数节点值拟合辅助点处近似函数的一阶导数，再以辅助点处近似函数的一阶导数值拟合节点处近似函数的二阶导数。由于在每一步中只需计算MLS形函数及其一阶导数，这种近似方法需要较少的影响点和较小的影响域。同时借助于Voronoi结构的优良几何性质，可以快速地搜索影响点。研究表明，与基于MLS的加权最小二乘无网格法（MWLS）相比，这种方法可以显著提高计算效率，并且在精度和收敛性方面也有所改善。

英文摘要：

In the point collocation-type meshless methods, evaluating the second order derivatives of the approximation functions is required, which is generally time consuming. In this paper, a Voronoi diagram based double grid diffuse （VDG） approximation is presented for fast evaluating the second order derivatives of the approximation functions. A meshless double grid least-square collocation method （MDGLS） is developed by discretizing PDEs with the weighted least-square method and approximating the trial functions with the proposed VDG approximation. In this method, the Voronoi diagram is constructed in the domain Ω by using nodes as generators, and then the Voronoi vertices are chosen as auxiliary points. The second order derivatives of the approximation functions are constructed in two steps. Firstly, the first order derivatives of the approximation functions at the auxiliary points are evaluated from the nodal value of the unknown functions at the nodes using the moving least square （MLS） approximation. Secondly, the second order derivatives are obtained at the nodes from the first order derivatives at auxiliary points using the MLS approximation. Only the first order derivatives of shape functions are evaluated in every step. As a consequence, the new method requires fewer support nodes and the smaller size of the domain of influence than those from MLS approximation. In addition, benefited from good properties of Voronoi structure, the influence points can be searched efficiently. Compared with the meshless weighted least-square.（MWLS） method, the proposed MDGLS method provides substantial improvement in computational efficiency.