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A Multiscale RBF Collocation Method for the Numerical Solution of Partial Differential Equations

Author

Listed:
  • Zhiyong Liu

    (School of Mathematical Sciences, Fudan University, Shanghai 200433, China
    school of Mathematics and Statistics, Ningxia University, Yinchuan 750021, China)

  • Qiuyan Xu

    (school of Mathematics and Statistics, Ningxia University, Yinchuan 750021, China)

Abstract

In this paper, we derive and discuss the hierarchical radial basis functions method for the approximation to Sobolev functions and the collocation to well-posed linear partial differential equations. Similar to multilevel splitting of finite element spaces, the hierarchical radial basis functions are constructed by employing successive refinement scattered data sets and scaled compactly supported radial basis functions with varying support radii. Compared with the compactly supported radial basis functions approximation and stationary multilevel approximation, the new method can not only solve the present problem on a single level with higher accuracy and lower computational cost, but also produce a highly sparse discrete algebraic system. These observations are obtained by taking the direct approach of numerical experimentation.

Suggested Citation

  • Zhiyong Liu & Qiuyan Xu, 2019. "A Multiscale RBF Collocation Method for the Numerical Solution of Partial Differential Equations," Mathematics, MDPI, vol. 7(10), pages 1-15, October.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:10:p:964-:d:276034
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    Citations

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    Cited by:

    1. Minghao Hu & Lihua Wang & Fan Yang & Yueting Zhou, 2023. "Weighted Radial Basis Collocation Method for the Nonlinear Inverse Helmholtz Problems," Mathematics, MDPI, vol. 11(3), pages 1-29, January.
    2. Wang, Rong & Xu, Qiuyan & Liu, Zhiyong & Yang, Jiye, 2023. "Fast meshfree methods for nonlinear radiation diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 438(C).
    3. Ju, Yuejuan & Yang, Jiye & Liu, Zhiyong & Xu, Qiuyan, 2023. "Meshfree methods for the variable-order fractional advection–diffusion equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 211(C), pages 489-514.
    4. Xu Xu & Jinyu Guo & Peixin Ye & Wenhui Zhang, 2023. "Approximation Properties of the Vector Weak Rescaled Pure Greedy Algorithm," Mathematics, MDPI, vol. 11(9), pages 1-23, April.

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