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Error analysis in Sobolev spaces for the improved moving least-square approximation and the improved element-free Galerkin method

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  • Li, Xiaolin
  • Chen, Hao
  • Wang, Yan

Abstract

The improved moving least-square (IMLS) approximation is a method to form shape functions in meshless methods. For the application of IMLS-based meshless methods to the numerical solution of boundary value problems, it is fundamental to analyze error of the IMLS approximation in Sobolev spaces. This paper begins by discussing properties of the IMLS shape function. Under appropriate assumption on weight functions, error estimates for the IMLS approximation are then established in Sobolev spaces in multiple dimensions. The improved element-free Galerkin (IEFG) method is a typical meshless Galerkin method based on coupling the IMLS approximation and Galerkin weak form. Error analysis of the IEFG method is also provided. Numerical examples are finally presented to prove the theoretical error results.

Suggested Citation

  • Li, Xiaolin & Chen, Hao & Wang, Yan, 2015. "Error analysis in Sobolev spaces for the improved moving least-square approximation and the improved element-free Galerkin method," Applied Mathematics and Computation, Elsevier, vol. 262(C), pages 56-78.
    • Handle: RePEc:eee:apmaco:v:262:y:2015:i:c:p:56-78
    DOI: 10.1016/j.amc.2015.04.002
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    References listed on IDEAS

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    1. Li, Xiaolin, 2011. "Development of a meshless Galerkin boundary node method for viscous fluid flows," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 82(2), pages 258-280.
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    Cited by:

    1. Shirzadi, Mohammad & Rostami, Mohammadreza & Dehghan, Mehdi & Li, Xiaolin, 2023. "American options pricing under regime-switching jump-diffusion models with meshfree finite point method," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
    2. Qu, Wenzhen & Sun, Linlin & Li, Po-Wei, 2021. "Bending analysis of simply supported and clamped thin elastic plates by using a modified version of the LMFS," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 185(C), pages 347-357.
    3. Zhijuan Meng & Xiaofei Chi & Lidong Ma, 2022. "A Hybrid Interpolating Meshless Method for 3D Advection–Diffusion Problems," Mathematics, MDPI, vol. 10(13), pages 1-21, June.
    4. Zhang, Tao & Li, Xiaolin, 2020. "Analysis of the element-free Galerkin method with penalty for general second-order elliptic problems," Applied Mathematics and Computation, Elsevier, vol. 380(C).
    5. Wang, Qiao & Zhou, Wei & Feng, Y.T. & Ma, Gang & Cheng, Yonggang & Chang, Xiaolin, 2019. "An adaptive orthogonal improved interpolating moving least-square method and a new boundary element-free method," Applied Mathematics and Computation, Elsevier, vol. 353(C), pages 347-370.
    6. Sun, FengXin & Wang, JuFeng, 2017. "Interpolating element-free Galerkin method for the regularized long wave equation and its error analysis," Applied Mathematics and Computation, Elsevier, vol. 315(C), pages 54-69.

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