Skip to main content
SpringerLink
Log in

Efficient Kansa-type MFS algorithm for elliptic problems

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this study we propose an efficient Kansa-type method of fundamental solutions (MFS-K) for the numerical solution of certain problems in circular geometries. In particular, we consider problems governed by the inhomogeneous Helmholtz equation in disks and annuli. The coefficient matrices in the linear systems resulting from the MFS-K discretization of these problems possess a block circulant structure and can thus be solved by means of a matrix decomposition algorithm and fast Fourier Transforms. Several numerical examples demonstrating the efficacy of the proposed algorithm are presented.

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. Alves, C.J.S.: On the choice of source points in the method of fundamental solutions. Eng. Anal. Bound. Elem. 33, 1348–1361 (2009)

    Article  MathSciNet  Google Scholar 

  2. Alves, C.J.S., Chen, C.S.: A new method of fundamental solutions applied to nonhomogeneous elliptic problems. Adv. Comput. Math. 23, 125–142 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  3. Alves, C.J.S., Valtchev, S.S.: A Kansa type method using fundamental solutions applied to elliptic PDEs. In: Leitão, V.M.A., Alves, C.J.S., Armando Duarte, C. (eds.) Advances in Meshfree Techniques, Computational Methods in Applied Sciences, pp. 241–256. Springer, Dordrecht (2007)

    Google Scholar 

  4. Bialecki, B., Fairweather, G.: Matrix decomposition algorithms for separable elliptic boundary value problems in two dimensions. J. Comput. Appl. Math. 46, 369–386 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  5. Davis, P.J.: Circulant Matrices. Wiley, New York (1979)

    MATH  Google Scholar 

  6. 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 

  7. Fairweather, G., Karageorghis, A., Martin, P.A.: The method of fundamental solutions for scattering and radiation problems. Eng. Anal. Bound. Elem. 27, 759–769 (2003)

    Article  MATH  Google Scholar 

  8. Golberg, M.A., Chen, C.S.: Discrete Projection Methods for Integral Equations. Computational Mechanics, Southampton (1997)

    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 and Mathematical Aspects, pp. 103–176. WIT/Computational Mechanics, Boston (1999)

    Google Scholar 

  10. Gorzelańczyk, P., Kołodziej, J.A.: Some remarks concerning the shape of the source contour with application of the method of fundamental solutions to elastic torsion of prismatic rods. Eng. Anal. Bound. Elem. 32, 64–75 (2008)

    Article  Google Scholar 

  11. Kansa, E.J.; Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics—II. Solutions to parabolic, hyperbolic and elliptic partial differential equations. Comput. Math. Appl. 19(8/9), 147–161 (1990)

    MATH  MathSciNet  Google Scholar 

  12. Karageorghis, A.: Efficient MFS algorithms in regular polygonal domains. Numer. Algorithms 50, 215–240 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Karageorghis, A.: A practical algorithm for determining the optimal pseudoboundary in the MFS. Adv. Appl. Math. Mech. 1, 510–528 (2009)

    MathSciNet  Google Scholar 

  14. Karageorghis, A., Chen, C.S., Smyrlis, Y.-S.: A matrix decomposition RBF algorithm: Approximation of functions and their derivatives. Appl. Numer. Math. 57, 304–319 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  15. Karageorghis, A., Chen, C.S., Smyrlis, Y.-S.: Matrix decomposition RBF algorithm for solving 3D elliptic problems. Eng. Anal. Bound. Elem. 33, 1368–1373 (2009)

    Article  MathSciNet  Google Scholar 

  16. Karageorghis, A., Smyrlis, Y.-S.: Matrix decomposition algorithms related to the MFS for axisymmetric problems. In: Manolis, G.D., Polyzos, D. (eds.) Recent Advances in Boundary Element Methods, pp. 223–237. Springer, New York (2009)

    Chapter  Google Scholar 

  17. The MathWorks: MATLAB. The MathWorks, Natick (2009)

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

    Article  MATH  MathSciNet  Google Scholar 

  19. Valtchev, S.S.: Numetical analysis of methods with fundamental solutions for acoustic and elastic wave propagation problems. PhD thesis, Department of Mathematics, Instituto Superior Téchnico, Universidade Técnica de Lisboa, Lisbon (2008)

  20. Valtchev, S.S., Roberty, N.C.: A time-marching MFS scheme for heat conduction problems. Eng. Anal. Bound. Elem. 32, 480–493 (2008)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, 1678, Nicosia, Cyprus

    Andreas Karageorghis

Authors
  1. Andreas Karageorghis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andreas Karageorghis.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Karageorghis, A. Efficient Kansa-type MFS algorithm for elliptic problems. Numer Algor 54, 261–278 (2010). https://doi.org/10.1007/s11075-009-9334-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-009-9334-8

Keywords

Mathematics Subject Classifications (2000)

Search

Navigation