In this paper, an adaptive element free Galerkin (EFG) method is presented to solve Poisson equation. In general, element free Galerkin method using moving least square (MLS) approximation needs a background mesh for integration. With the arbitrary polygonal influence domain technique, the shape function of MLS has almost interpolation property, and the Gaussian quadrature points in the background integration element only contribute to the vertices of that element, which enables us to compute the residual based on the background integration element just like the finite element method (FEM). The adaptive procedure based on triangular or tetrahedral background integration elements is then developed for EFG method, in which the residual-based a posteriori error estimation of FEM is used. Numerical examples are provided to illustrate the efficiency of the proposed adaptive EFG method.

本文提出了一种自适应无元伽辽金(EFG)方法来求解泊松方程。通常，使用移动最小二乘 (MLS) 近似的无单元 Galerkin 方法需要背景网格进行积分。使用任意多边形影响域技术，MLS的形状函数几乎具有插值性质，背景积分元素中的高斯正交点仅对该元素的顶点有贡献，这使我们能够基于背景积分元素计算残差就像有限元法（FEM）一样。然后为 EFG 方法开发了基于三角形或四面体背景积分元素的自适应程序，其中基于残差的后验使用 FEM 的误差估计。提供了数值例子来说明所提出的自适应 EFG 方法的效率。