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# A Dimension-Splitting Variational Multiscale Element-Free Galerkin Method for Three-Dimensional Singularly Perturbed Convection-Diffusion Problems

Jufeng Wang1, Yong Wu1, Ying Xu1, Fengxin Sun2,*

1 College of Finance & Information, Ningbo University of Finance & Economics, Ningbo, 315175, China
2 Faculty of Science, Ningbo University of Technology, Ningbo, 315016, China

* Corresponding Author: Fengxin Sun. Email:

(This article belongs to this Special Issue: Numerical Methods in Engineering Analysis, Data Analysis and Artificial Intelligence)

Computer Modeling in Engineering & Sciences 2023, 135(1), 341-356. https://doi.org/10.32604/cmes.2022.023140

## Abstract

By introducing the dimensional splitting (DS) method into the multiscale interpolating element-free Galerkin (VMIEFG) method, a dimension-splitting multiscale interpolating element-free Galerkin (DS-VMIEFG) method is proposed for three-dimensional (3D) singular perturbed convection-diffusion (SPCD) problems. In the DS-VMIEFG method, the 3D problem is decomposed into a series of 2D problems by the DS method, and the discrete equations on the 2D splitting surface are obtained by the VMIEFG method. The improved interpolation-type moving least squares (IIMLS) method is used to construct shape functions in the weak form and to combine 2D discrete equations into a global system of discrete equations for the three-dimensional SPCD problems. The solved numerical example verifies the effectiveness of the method in this paper for the 3D SPCD problems. The numerical solution will gradually converge to the analytical solution with the increase in the number of nodes. For extremely small singular diffusion coefficients, the numerical solution will avoid numerical oscillation and has high computational stability.

## Keywords

Wang, J., Wu, Y., Xu, Y., Sun, F. (2023). A Dimension-Splitting Variational Multiscale Element-Free Galerkin Method for Three-Dimensional Singularly Perturbed Convection-Diffusion Problems. CMES-Computer Modeling in Engineering & Sciences, 135(1), 341–356.

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