For very strong convection-dominated problems, stabilized meshless methods such as variational multiscale element-free Galerkin (VMEFG) method may still produce over- and under-shootings near the boundary or interior layers. In this paper, an adaptive VMEFG method is presented to solve convection–diffusion equations with convection-dominated. The adaptive algorithm based on background integration cell locates high gradient region with Zienkiewicz–Zhu indicator and refine the nodes in the region to improve the computational accuracy of VMEFG method. Meanwhile, this adaptive algorithm can also be used in element-free Galerkin (EFG) method. To compare and verify the validity of the proposed adaptive VMEFG method in convection-dominated problem, seven case studies are calculated by the adaptive VMEFG and EFG methods. The numerical experiments show that the proposed adaptive algorithm can not only refine the singularity regions well, but also is simple, effective and efficient for convection-dominated problem.

对于非常强的对流为主的问题，稳定的无网格方法如变分多尺度无单元伽辽金 (VMEFG) 方法可能仍然会在边界或内部层附近产生过冲和欠冲。在本文中，提出了一种自适应 VMEFG 方法来求解对流主导的对流-扩散方程。基于背景积分单元的自适应算法通过Zienkiewicz-Zhu指标定位高梯度区域并细化该区域的节点以提高VMEFG方法的计算精度。同时，该自适应算法也可用于无元素伽辽金（EFG）方法。为了比较和验证所提出的自适应 VMEFG 方法在对流主导问题中的有效性，通过自适应 VMEFG 和 EFG 方法计算了七个案例研究。