Advances in the Element-Free Galerkin Method: From Linear Solid Mechanics to Multi-Physics Applications and Hybrid Domain Coupling

The Element-Free Galerkin (EFG) method was originally developed for linear solid mechanics problems, using Moving Least Squares (MLS) approximations to construct shape functions for the numerical approximation of the displacement field and its variations within the weak form of the equilibrium equations. Over the past decades, it has evolved into a versatile meshfree framework applicable to a broad spectrum of engineering and scientific problems. This review provides a comprehensive account of the main advances in EFG, tracing its development from the original formulation and early challenges to the strategies devised to overcome them. Subsequent improvements in accuracy, stability, and computational efficiency are examined in detail, together with alternative shape function constructions such as Moving Kriging (MK) and Local Maximum Entropy (LME) approximations. The extension of EFG to multiphysics problems is discussed, emphasizing how analogies with the Finite Element Method (FEM) have enabled the adaptation of established stabilization and enrichment techniques. Hybrid FEM–EFG coupling strategies are also reviewed. The article concludes with a survey of significant applications in mechanics and transport phenomena, highlighting their broader implications in science and technology.