Advertisement

An Economic Method for Evaluation of Volume Integrals

  • Natalia T. Kolkovska
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3401)

Abstract

Approximations to the volume integrals with logarithmic kernels are obtained as solutions to finite difference schemes on the regular grid in a rectangular domain, which contains the domain of integration. Two types of right-hand sides in the finite difference equations are constructed – the first one includes some double and line integrals and the second one is fully discrete. The error estimates (optimal with respect to the smoothness of the volume integral) for both types of right-hand sides are obtained for volume integrals in appropriate Besov spaces.

Keywords

volume integrals integral equations particular solution Poisson equation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Besov, O., Il’in, V., Nikolskii, S.: Integral representations of functions and imbedding theorems. John Wiley & Sons, New York (1979)zbMATHGoogle Scholar
  2. 2.
    Ethridge, F., Greengard, L.: A new fast-multipole accelerated Poisson solver in two dimensions. Siam J. Sci. Comp. 23, 741–760 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Friedman, A.: Mathematics in industrial problems. part 5. Springer, Heidelberg (1992)zbMATHGoogle Scholar
  4. 4.
    Greenbaum, A., Mayo, A.: Rapid parallel evaluation of integrals in potential theory on general three-dimensional regions. J. Comput. Physics 145, 731–742 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Kolkovska, N.: Numerical methods for computation of the double layer logarithmic potential. In: Vulkov, L.G., Wasniewski, J., Yalamov, P.Y. (eds.) WNAA 1996. LNCS, vol. 1196, pp. 243–249. Springer, Heidelberg (1997)Google Scholar
  6. 6.
    Kolkovska, N.: A finite difference scheme for computation of the logarithmic potential. In: Finite difference methods, Theory and Applications, pp. 139–144. Nova Science Publishers, Bombay (1999)Google Scholar
  7. 7.
    Lazarov, R., Mokin, Y.: On the computation of the logarithmic potential. Soviet Math. Dokl. 272, 320–323 (1983)MathSciNetGoogle Scholar
  8. 8.
    Mayo, A.: The fast solution of Poisson’s and the biharmonic equations on irregular regions. SIAM J. Numer. Anal. 21, 285–299 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Mayo, A.: The rapid evaluation of volume integrals of potential theory on general regions. J. Comput. Physics 100, 236–245 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    McKenney, A., Greengard, L., Mayo, A.: A fast Poisson solver for complex geometries. J. Comp. Physics 118, 348–355 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Samarskii, A., Lazarov, R., Makarov, V.: Difference schemes for differential equations having generalized solutions. Vysshaya Shkola Publishers, Moscow (1987) (in Russian)Google Scholar
  12. 12.
    Triebel, H.: Theory of function spaces. Birkhauser, Basel (1983)CrossRefGoogle Scholar
  13. 13.
    Vladimirov, V.: Equations in mathematical physics. Moscow, Nauka (1976) (in Russian)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Natalia T. Kolkovska
    • 1
  1. 1.Institute of Mathematics and InformaticsBulgarian Academy of Sciences 

Personalised recommendations