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A Meshless Finite Difference Method Based on Polynomial Interpolation

  • X. W. HuangEmail author
  • C. S. Wu
Article
  • 62 Downloads

Abstract

The finite difference (FD) formula plays an important role in the meshless methods for the numerical solution of partial differential equations. It can be created by polynomial interpolation, however, this idea has not been widely used due to the complexity of multivariate polynomial interpolation. Instead, radial basis functions interpolation is widely used to generate the FD formula. In this paper, we first propose a simple and practicable node distribution, which makes it convenient for one to choose the interpolation node set that guarantee the unique solvability of multivariate polynomial interpolation. The greatest advantage of the interpolation node set is that we can only face triangular matrix in order to obtain the Lagrange basis polynomials by the constructed basis polynomials. We then use Taylor’s formula to establish error estimates of the FD formula based on polynomial interpolation. We finally give some numerical experiments for the numerical solutions of the Poisson equation and the heat equation.

Keywords

Finite difference Meshless methods Multivariate polynomial interpolation 

Mathematics Subject Classification

65D25 41A10 41A63 65N06 

Notes

References

  1. 1.
    Barnett, G.A.: A robust RBF-FD formulation based on polyharmonic splines and polynomials. Ph.D. Thesis, the Department of Applied Mathematics at the University of Colorado, Boulder (2015)Google Scholar
  2. 2.
    Bayona, V., Fornberg, B., Flyer, N., Barnett, G.A.: On the role of polynomials in RBF-FD approximations: II. Numerical solution of elliptic PDEs. J. Comput. Phys. 332, 257–273 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bayona, V., Moscoso, M., Carretero, M., Kindelan, M.: RBF-FD formulas and convergence properties. J. Comput. Phys. 229(22), 8281–8295 (2010)CrossRefzbMATHGoogle Scholar
  4. 4.
    Cao, Y., Dong, J.L., Yao, L.Q.: A modification of the moving least-squares approximation in the element-free Galerkin method. J. Appl. Math. 2014(2), 1–13 (2014)zbMATHGoogle Scholar
  5. 5.
    Davydov, O., Dang, T.O.: On the optimal shape parameter for Gaussian radial basis function finite difference approximation of the Poisson equation. Comput. Math. Appl. 62(5), 2143–2161 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Fasshauer, G.E., Mccourt, M.J.: Stable evaluation of Gaussian radial basis function interpolants. SIAM J. Sci. Comput. 34(2), 737–762 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Flyer, N., Barnett, G.A., Wicker, L.J.: Enhancing finite differences with radial basis functions: experiments on the Navier–Stokes equations. J. Comput. Phys. 316, 39–62 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Flyer, N., Fornberg, B., Bayona, V., Barnett, G.A.: On the role of polynomials in RBF-FD approximations: I. Interpolation and accuracy. J. Comput. Phys. 321, 21–38 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fornberg, B., Flyer, N.: Fast generation of 2-D node distributions for mesh-free PDE discretizations. Comput. Math. Appl. 69(7), 531–544 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fornberg, B., Flyer, N.: Solving PDEs with radial basis functions. Acta Numer. 24, 215–258 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Fornberg, B., Larsson, E., Flyer, N.: Stable computations with Gaussian radial basis functions. SIAM J. Sci. Comput. 33(2), 869–892 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fornberg, B., Lehto, E., Powell, C.: Stable calculation of Gaussian-based RBF-FD stencils. Comput. Math. Appl. 65(4), 627–637 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gasca, M., Maeztu, J.I.: On Lagrange and Hermite interpolation in \({R}^k\). Numer. Math. 39(1), 1–14 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gasca, M., Sauer, T.: Polynomial interpolation in several variables. Adv. Comput. Math. 12(4), 377–410 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lancaster, P., Salkauskas, K.: Surfaces generated by moving least squares methods. Math. Comput. 37(155), 141–158 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Larsson, E., Fornberg, B.: Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions. Comput. Math. Appl. 49(1), 103–130 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Larsson, E., Lehto, E., Heryudono, A., Fornberg, B.: Stable computation of differentiation matrices and scattered node stencils based on Gaussian radial basis functions. SIAM J. Sci. Comput. 35(4), A2096–A2119 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lemeire, F.: Bounds for condition numbers of triangular and trapezoid matrices. Bit Numer. Math. 15(1), 58–64 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Li, X.: Error estimates for the moving least-square approximation and the element-free Galerkin method in n-dimensional spaces. Appl. Numer. Math. 99(C), 77–97 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Li, X., Chen, H., Wang, Y.: Error analysis in Sobolev spaces for the improved moving least-square approximation and the improved element-free Galerkin method. Appl. Math. Comput. 262, 56–78 (2015)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Liang, X.Z., Lü, C.M., Feng, R.Z.: Properly posed sets of nodes for multivariate Lagrange interpolation in \({C}^s\). SIAM J. Numer. Anal. 39(2), 587–595 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Sauer, T., Xu, Y.: On multivariate Lagrange interpolation. Math. Comput. 64(211), 1147–1170 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Shadrin, A.: Error bounds for Lagrange interpolation. J. Approx. Theory 80(1), 25–49 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Shankar, V.: The overlapped radial basis function-finite difference (RBF-FD) method: a generalization of RBF-FD. J. Comput. Phys. 342, 211–228 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Shankar, V., Fogelson, A.L.: Hyperviscosity-based stabilization for radial basis function-finite difference (RBF-FD) discretizations of advection–diffusion equations. J. Comput. Phys. 372, 616–639 (2018)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Shu, C.: Differential Quadrature and Its Application in Engineering. Springer, London (2000)CrossRefzbMATHGoogle Scholar
  27. 27.
    Shu, C., Yao, Q., Yeo, K.S.: Block-marching in time with DQ discretization: an efficient method for time-dependent problems. Comput. Methods Appl. Mech. Eng. 191(41), 4587–4597 (2002)CrossRefzbMATHGoogle Scholar
  28. 28.
    Sroka, G.: Constants in V.A. Markov’s inequality in \({L}_p\) norms. J. Approx. Theory 194(C), 27–34 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Wang, H.Y., Cui, F., Wang, X.H.: Explicit representations for local Lagrangian numerical differentiation. Acta Math. Sin. Engl. Ser. 23(2), 365–372 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Wang, X.H., Cui, F.: Stable Lagrangian numerical differentiation with the highest order of approximation. Sci. China Ser. A Math. 49(2), 225–232 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2005)zbMATHGoogle Scholar
  32. 32.
    Whitney, H.: Functions differentiable on the boundaries of regions. Ann. Math. 35(3), 482–485 (1934)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Wright, G., Fornberg, B.: Scattered node compact finite difference-type formulas generated from radial basis functions. J. Comput. Phys. 212(1), 99–123 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of SciencesWuhan University of TechnologyWuhanChina

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