Skip to main content
SpringerLink
Log in

Convergence of the method of fundamental solutions for Poisson’s equation on the unit sphere

  • Selected abstratcs
  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

Solutions of boundary value problems of the Laplace equation on the unit sphere are constructed by using the fundamental solution

$$\Phi (\bf{x},\bf{y})=\frac{1}{4\pi \|\bf{x}-\bf{y}\|},\qquad \bf{x}, \bf{y}\in R^3.$$

With the use of radial basis approximation for finding particular solutions of Poisson's equation, the rate of convergence of the method of fundamental solutions is derived for solving the boundary value problems of Poisson’s equation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Atkinson, K.E.: The numerical evaluation of particular solutions for Poisson’s equation. IMA J. Numer. Anal. 5, 319–338 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  2. Berens, H., Li, L.Q.: On the de la Vallée–Poussin means on the sphere. Results Math. 24, 12–26 (1993)

    MATH  MathSciNet  Google Scholar 

  3. Bogomolny, A.: Fundamental solutions method for elliptic boundary value problems. SIAM J. Numer. Anal. 22(4), 644–669 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  4. Cessenat, O., Després, B.: Using plane waves as base functions for solving time harmonic equations with the ultra weak variational formulation. J. Comput. Acoust. 11(2), 227–238 (2003)

    Article  MathSciNet  Google Scholar 

  5. Cheng, R.S.C.: Delta-Trigonometric and spline methods using the single-layer potential representation. PhD dissertation, University of Maryland (1987)

  6. Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory. Springer, Berlin Heidelberg New York (1992)

    MATH  Google Scholar 

  7. Fairweather, G., Karageorghis, A.: The method of fundamental solutions for elliptic boundary value problems. Adv. Comput. Math. 9, 69–95 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  8. Golberg, M.A., Chen, C.S.: Discrete Projection Methods for Integral Equations. Computational Mechanics, Southampton (1996)

    Google Scholar 

  9. Golberg, M.A., Chen, C.S.: The method of fundamental solutions for potential, Helmholtz and diffusion problems. In: Golberg, M.A. (ed.) Boundary Integral Methods – Numerical and Mathematical Aspects, Computational Mechanics, Southampton, pp 103–176 (1998)

  10. Greengard, L., Rokhlin, V.: A new version of the fast multipole method for the Laplace equation in three dimensions. Acta Numer., 229–269 (1997)

  11. Herrera, I.: Trefftz method: A general theory. Numer. Methods Partial Differ. Equ. 16(6), 561–580 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  12. Hussaini, M.Y., Kopriva, D.A., Patera, A.T.: Spectral collocation methods. Appl. Numer. Math. 5, 177–208 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  13. Katsurada, M.: A mathematical study of the charge simulation method II. J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 36, 135–162 (1989)

    MATH  MathSciNet  Google Scholar 

  14. Katsurada, M.: Asymptotic error analysis of the charge simulation method in a Jordan region with an analytic boundary. J. Fac. Sci., Univ. Tokyo, Sec. 1A, Math. 37, 635–657 (1990)

    MATH  MathSciNet  Google Scholar 

  15. Katsurada, M.: Charge simulation method using exterior mapping functions. Jpn. J. Ind. Appl. Math. 11, 47–61 (1994)

    MATH  MathSciNet  Google Scholar 

  16. Katsurada, M., Okamoto, H.: A mathematical study of the charge simulation method I. J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 35, 507–518 (1988)

    MATH  MathSciNet  Google Scholar 

  17. Katsurada, M., Okamoto, H.: The collocation points of the fundamental solution method for the potential problem. Comput. Math. Appl. 31, 123–137 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kitagawa, T.: On the numerical stability of the method of fundamental solution applied to the Dirichlet problem. Jpn. J. Appl. Math. 5, 123–133 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  19. Li, X.: On convergence of the method of fundamental solutions for solving the Dirichlet problem of Poisson's equation. Adv. Comput. Math. 23, 265–277 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  20. Li, X.: Convergence of the method of fundamental solutions for solving the boundary value problem of modified Helmholtz equation. Appl. Math. Comput. 159, 113–125 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  21. Li, X., Golberg, M.A.: On the convergence of the dual reciprocity method for Poisson's equation. In: Rashed, Y.F., Brebbia, C.A. (eds.) Transformation of Domain Effects to the Boundary, pp. 227–251. WIT, Boston (2003)

    Google Scholar 

  22. Li, X., Ho, C.H., Chen, C.S.: Computational test of approximation of functions and their derivatives by radial basis functions. Neural Parallel Sci. Comput. 10, 25–46 (2002)

    MATH  MathSciNet  Google Scholar 

  23. Li, X., Micchelli, C.A.: Approximation by radial bases and neural networks. Numer. Algorithms 25, 241–262 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  24. Miranda, C.: Partial Differential Equations of Elliptic Type. Springer, Berlin Heidelberg New York (1970)

    MATH  Google Scholar 

  25. Müller, C.: Analysis of Spherical Symmetries in Euclidean Spaces. Springer, Berlin Heidelberg New York (1998)

    MATH  Google Scholar 

  26. Smyrlis, Y.S., Karageorghis, A.: Some aspects of the method of fundamental solutions for certain harmonic problems. J. Sci. Comput. 16(3), 341–371 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  27. Smyrlis, Y.S., Karageorghis, A.: Numerical analysis of the MFS for certain harmonic problems. Modél. Math. Anal. Numér. 38, 495–517 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  28. Tsangaris, T., Smyrlis, Y.S., Karageoghis, A.: Numerical analysis of the MFS for harmonic problems in annular domains. Numer. Methods Partial Differ. Equ. 22(3), 507–539 (2006)

  29. Ushijima, T., Chiba, F.: A fundamental solution method for the reduced wave problem in a domain exterior to a disc. J. Comput. Appl. Math. 152, 545–557 (2003)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Nevada, Las Vegas, NV, 89154-4020, USA

    Xin Li

Authors
  1. Xin Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xin Li.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, X. Convergence of the method of fundamental solutions for Poisson’s equation on the unit sphere. Adv Comput Math 28, 269–282 (2008). https://doi.org/10.1007/s10444-006-9022-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-006-9022-3

Keywords

Mathematics Subject Classifications (2000)

Search

Navigation