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Analysis of Meshfree Galerkin Methods Based on Moving Least Squares and Local Maximum-Entropy Approximation Schemes

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Listed:
  • Hongtao Yang

    (College of Engineering, Peking University, Beijing 100187, China)

  • Hao Wang

    (School of Mechanical Engineering, Purdue University, 585 Purdue Mall, West Lafayette, IN 47907, USA)

  • Bo Li

    (College of Engineering, Peking University, Beijing 100187, China)

Abstract

Over the last two decades, meshfree Galerkin methods have become increasingly popular in solid and fluid mechanics applications. A variety of these methods have been developed, each incorporating unique meshfree approximation schemes to enhance their performance. In this study, we examine the application of the Moving Least Squares and Local Maximum-Entropy (LME) approximations within the framework of Optimal Transportation Meshfree for solving Galerkin boundary-value problems. We focus on how the choice of basis order and the non-negativity, as well as the weak Kronecker-delta properties of shape functions, influence the performance of numerical solutions. Through comparative numerical experiments, we evaluate the efficiency, accuracy, and capabilities of these two approximation schemes. The decision to use one method over the other often hinges on factors like computational efficiency and resource management, underscoring the importance of carefully considering the specific attributes of the data and the intrinsic nature of the problem being addressed.

Suggested Citation

  • Hongtao Yang & Hao Wang & Bo Li, 2024. "Analysis of Meshfree Galerkin Methods Based on Moving Least Squares and Local Maximum-Entropy Approximation Schemes," Mathematics, MDPI, vol. 12(3), pages 1-20, February.
  • Handle: RePEc:gam:jmathe:v:12:y:2024:i:3:p:494-:d:1333164
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    References listed on IDEAS

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    1. Joldes, Grand Roman & Chowdhury, Habibullah Amin & Wittek, Adam & Doyle, Barry & Miller, Karol, 2015. "Modified moving least squares with polynomial bases for scattered data approximation," Applied Mathematics and Computation, Elsevier, vol. 266(C), pages 893-902.
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