无网格法是一类重要的科学工程数值方法，能克服网格类方法如有限元法、差分法等对网格单元的依赖。无单元Galerkin法是一种常用的无网格法，本项目将针对该方法分析三维边值问题时计算量大的缺点，研究三维复变量移动最小二乘近似的算法和误差估计理论，建立三维问题的复变量无单元Galerkin法及其误差估计理论，降低相关无网格法的计算量。然后，针对无网格法中常用的Gauss积分法在数值积分时计算量大的缺点，建立适用于三维问题和不可压缩流体问题的高效稳定的嵌套子域积分法，并理论分析积分误差，进一步显著降低相关无网格法的计算量。最后，基于三维复变量移动最小二乘近似和嵌套子域积分法，研究求解不可压缩Stokes方程和Navier-Stokes方程的高效稳定无单元Galerkin法，并理论分析稳定性和误差。本项目将为分析不可压缩流体力学方程提供高效稳定的无网格算法，促进流体力学无网格法的研究进展。

Meshless (or meshfree) method, which is an important numerical method in science and engineering, can overcome the meshing-related difficulties involved in some traditional numerical methods such as finite element method and finite difference method. For the shortcoming of the costly computation of the element-free Galerkin method, which is an often-used meshless method, for three-dimensional problems, computational formulas and theoretical error estimates of the three-dimensional complex variable moving least squares approximation will be researched in this project, and the complex variable element-free Galerkin method and its theoretical error estimates will also be discussed for three-dimensional problems. The complex variable element-free Galerkin method will reduce the computational cost of the element-free Galerkin method. Then, for the shortcoming of the costly computation of the often-used Gauss integration in meshless method, an efficient and stable nesting sub-domain integration algorithm for three-dimensional problems and incompressible fluid problems will be presented to further reduce the computational cost. The theoretical error estimates of the integration algorithm will be discussed. At last, using the three-dimensional complex variable moving least squares approximation and the nesting sub-domain integration algorithm, an efficient and stable element-free Galerkin method and its stability and error will be presented for the incompressible Stokes and Navier-Stokes equations. This project will provide efficient and stable meshless algorithms for numerical analysis of incompressible fluid problems and will advance the development of meshless methods for fluid problems.