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An adaptive variational multiscale element free Galerkin method for convection–diffusion equations

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Abstract

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.

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Acknowledgements

This work is financially supported by the fund of Hubei International Science and Technology Cooperation Base of Fish Passage (no. HIBF2020006).

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Authors and Affiliations

  1. College of Science and Hubei International Science and Technology Cooperation Base of Fish Passage, China Three Gorges University, Yichang, 443002, China

    Xiaohua Zhang

  2. College of Science and Three Gorges Math Research Center, China Three Gorges University, Yichang, 443002, China

    Ping Zhang & Wenjie Qin

  3. Hubei International Science and Technology Cooperation Base of Fish Passage, China Three Gorges University, Yichang, 443002, China

    Xiaotao Shi

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  1. Xiaohua Zhang
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Correspondence to Xiaohua Zhang.

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Zhang, X., Zhang, P., Qin, W. et al. An adaptive variational multiscale element free Galerkin method for convection–diffusion equations. Engineering with Computers 38 (Suppl 4), 3373–3390 (2022). https://doi.org/10.1007/s00366-021-01469-6

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  • DOI: https://doi.org/10.1007/s00366-021-01469-6

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