Summary In this paper, by combining the dimension splitting method and the improved complex variable element‐free Galerkin method, the dimension splitting and improved complex variable element‐free Galerkin (DS‐ICVEFG) method is presented for 3‐dimensional (3D) transient heat conduction problems. Using the dimension splitting method, a 3D transient heat conduction problem is translated into a series of 2‐dimensional ones, which can be solved with the improved complex variable element‐free Galerkin (ICVEFG) method. In the ICVEFG method for each 2‐dimensional problem, the improved complex variable moving least‐square approximation is used to obtain the shape functions, and the penalty method is used to apply the essential boundary conditions. Finite difference method is used in the 1‐dimensional direction, and the Galerkin weak form of 3D transient heat conduction problem is used to obtain the final discretized equations. Then, the DS‐ICVEFG method for 3D transient heat conduction problems is presented. Four numerical examples are given to show that the new method has higher computational precision and efficiency.

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This paper presents a hybrid element-free Galerkin (HEFG) method for solving wave propagation problems. By introducing the dimension split method, the three-dimensional wave propagation problems are transformed into a series of two-dimensional ones in other one-dimensional directions. The two-dimensional problems are solved using the improved element-free Galerkin (IEFG) method, and the finite difference method is used in the one-dimensional splitting direction and the time space. Then, the formulas of the HEFG method for three-dimensional wave propagation problems are obtained. Numerical examples are selected to show the effectiveness and the advantage of the HEFG method. The convergence and error analysis of the HEFG method are discussed according to the numerical results under different splitting directions, weight functions, node distributions, scale parameters of the influence domain, penalty factors, and time steps. The numerical results are given to show the convergence and advantages of the HEFG method over the IEFG method. Comparing with the IEFG method, the HEFG method has greater computational precision and speed for three-dimensional wave propagation problems. KEYWORDS dimension split method, finite difference method, hybrid element-free Galerkin method, improved element-free Galerkin method, improved moving least squares approximation, wave propagation problem Int J Numer Methods Eng. 2019;117:15-37.wileyonlinelibrary.com/journal/nme