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Octant-Based Stencil Selection for Meshless Finite Difference Methods in 3D

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Abstract

We study a simple meshless stencil selection algorithm in 3D for supporting the meshless finite difference method based on radial basis functions (RBF-FD) to solve the Dirichlet problem for the Poisson equation. Our numerical experiments show that the proposed method produces solutions that differ very little from the solutions by the finite element method with Courant’s piecewise linear basis functions, when the domain is discretized by the vertices of the respective tetrahedral triangulation.

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References

  1. Buhmann, M.D.: Radial Basis Functions. Cambridge University Press, New York (2003)

    Book  Google Scholar 

  2. Davydov, O., Oanh, D.T.: Adaptive meshless centres and RBF stencils for Poisson equation. J. Comput. Phys. 230, 287–304 (2011)

    Article  MathSciNet  Google Scholar 

  3. Davydov, O., Oanh, D.T.: On the optimal shape parameter for Gaussian radial basis function finite difference approximation of the Poisson equation. Comput. Math. Appl. 62, 2143–2161 (2011)

    Article  MathSciNet  Google Scholar 

  4. Davydov, O., Schaback, R.: Error bounds for kernel-based numerical differentiation. Numer. Math. 132, 243–269 (2016)

    Article  MathSciNet  Google Scholar 

  5. Davydov, O., Schaback, R.: Minimal numerical differentiation formulas. Numer. Math. 140, 555–592 (2018)

    Article  MathSciNet  Google Scholar 

  6. Davydov, O., Schaback, R.: Optimal stencils in Sobolev spaces. IMA J. Numer. Anal. 39, 398–422 (2019)

    MathSciNet  Google Scholar 

  7. Fasshauer, G.F.: Meshfree Approximation Methods with MATLAB. World Scientific Publishing Co., Inc., River Edge (2007)

    Book  Google Scholar 

  8. Flyer, N., Fornberg, B., Bayona, V., Barnett, G.A.: On the role of polynomials in RBF-FD approximations: i. i.terpolation and accuracy. J. Comput. Phys. 321, 21–38 (2016)

    Article  MathSciNet  Google Scholar 

  9. Fornberg, B., Flyer, N.: A Primer on Radial Basis Functions with Applications to the Geosciences. Society for Industrial and Applied Mathematics, Philadelphia, PA (2015)

  10. Fornberg, B., Larsson, E., Flyer, N.: Stable computations with Gaussian radial basis functions. SIAM J. Sci. Comput. 33, 869–892 (2011)

    Article  MathSciNet  Google Scholar 

  11. Liszka, T., Orkisz, J.: The finite difference method at arbitrary irregular grids and its application in applied mechanics. Comput. Struct. 11, 83–95 (1980)

    Article  MathSciNet  Google Scholar 

  12. MathWorks: Inc: partial differential equation ToolboxTM User?s Guide. The MathWorks, Inc (2016)

  13. Oanh, D.T., Davydov, O., Phu, H.X.: Adaptive RBF-FD method for elliptic problems with point singularities in 2D. Appl. Math. Comput. 313, 474–497 (2017)

    MathSciNet  MATH  Google Scholar 

  14. Ureña, M., Benito, J.J., Ureña, F., García, Á., Gavete, L., Benito, L.: Adaptive strategies to improve the application of the generalized finite differences method in 2D and 3D. Math. Methods Appl. Sci. 41, 7115–7129 (2018)

    Article  MathSciNet  Google Scholar 

  15. Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2005)

    MATH  Google Scholar 

Download references

Acknowledgements

This paper was supported by Thai Nguyen University under grant number DH2015-TN07-03. We are grateful to three anonymous referees for their useful suggestions that have helped to improve our paper.

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Authors and Affiliations

  1. Department of Mathematics, University of Giessen, Arndtstrasse 2, 35392, Giessen, Germany

    Oleg Davydov

  2. Division of Science-Technology & International Cooperation, Thai Nguyen University of Information & Communication Technology, Quyet Thang Ward, Thai Nguyen City, Vietnam

    Dang Thi Oanh

  3. Department of Basic Sciences, Thai Nguyen University of Information & Communication Technology, Quyet Thang Ward, Thai Nguyen City, Vietnam

    Ngo Manh Tuong

Authors
  1. Oleg Davydov
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  2. Dang Thi Oanh
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  3. Ngo Manh Tuong
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Correspondence to Dang Thi Oanh.

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Davydov, O., Oanh, D.T. & Tuong, N.M. Octant-Based Stencil Selection for Meshless Finite Difference Methods in 3D. Vietnam J. Math. 48, 93–106 (2020). https://doi.org/10.1007/s10013-019-00364-4

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  • DOI: https://doi.org/10.1007/s10013-019-00364-4

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