Development, applications and prospects of the element-free Galerkin method

YanYuJie ZHENG, Ruo WANG, ChangMin FU, WenShan HU

Prog Geophy ›› 2026, Vol. 41 ›› Issue (5) : 2012-2025.

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Prog Geophy ›› 2026, Vol. 41 ›› Issue (5) : 2012-2025. DOI: 10.6038/pg2026JJ0313

Development, applications and prospects of the element-free Galerkin method

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Abstract

Numerical simulation (forward modeling) serves as a core tool for analyzing complex physical processes and plays an irreplaceable role in multiple fields such as geophysics. However, traditional grid-based methods like the finite element method, despite their wide application due to flexible grid discretization, encounter difficulties in handling irregular geometries—especially those with sharp corners, the problem has been resolved by meshless methods, which only require the deployment of nodes without the need for mesh division. The element free Galerkin method, developed in the past two to three decades, holds a prominent position among meshless methods due to its rigorous theoretical foundation and algorithmic stability, and demonstrats remarkable advantages in numerical simulation. The shape functions constructed through moving least squares approximation inherently possess high-order continuity, thereby enhancing simulation accuracy. When simulating models with sharp inner angles, holes, and other complex boundaries, the element free Galerkin method does not rely on grid reconstruction, fundamentally eliminating the computational bias caused by grid distortion. This paper systematically reviews the theoretical framework, development history, application fields of the element free Galerkin method, and summarizes the differences in accuracy and computational efficiency among the Element-Free Galerkin Method, the Finite Element Method and the Finite Difference Method in their applications to the field of geophysics. Through this comprehensive review, it provides comprehensive materials for deeper research and engineering practice of the element free Galerkin method, with the aim of promoting its continuous development.

Key words

Element-free Galerkin method / Moving Least Square method / Numerical simulation / Geophysics

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YanYuJie ZHENG , Ruo WANG , ChangMin FU , et al. Development, applications and prospects of the element-free Galerkin method[J]. Progress in Geophysics. 2026, 41(5): 2012-2025 https://doi.org/10.6038/pg2026JJ0313

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