Boundary collocation fast poisson solver on irregular domains

  • Daeshik Lee


A fast Poisson solver on irregular domains, based on boundary methods, is presented. The harmonic polynomial approximation of the solution of the associated homogeneous problem provides a good practical boundary method which allows a trivial parallel processing for solution evaluation or straightforward computations of the interface values for domain decomposition/embedding.

AMS Mathematics Subject Classification

65N35 65N55 65Y05 

Key words and pharases

harmonic polynomial approximation fast Poisson solver irregular domain domain decomposition/embedding 


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  1. 1.
    A. Bogomolny,Fundamental solutions method for elliptic boundary value problems, SIAM J. Numer. Anal., 22 (1985), pp. 644.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    A. Brandt,Multilevel computations: review and recent developments, Proc. 3d Copper Mountain Conf. on multigrid methods, 1989.Google Scholar
  3. 3.
    B. Buzbee, F. Dorr, A. George andG. Golub,The direct solution of the discrete Poisson equation on irregular regions, SIAM J. Numer. Anal., 8 (1971), pp. 722–736.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    C. Canuto, M. Hussaini, A. Quarteroni and T. Zang,Spectral Methods in Fluid Dynamics, S-V, 1988.Google Scholar
  5. 5.
    P. Concus, G. Golub and D. O’Leary,A generalized conjugate gradient method for the numerical solution of partial differential equations, in J. Bunch and D. Rose, ed., Proc. Symp. on Sparse Matrix Computations, Acad. Press, 1975, pp. 309–332.Google Scholar
  6. 6.
    R. Dautray andJ.-L. Lions,Mathematical Analysis and Numerical Methods for Science and Technology, vol. 1, S-V, 1990.Google Scholar
  7. 7.
    M. Dryja and O. Widlund,Some domain decomposition algorithms for elliptic problems, in Iterative Methods for Large Linear Systems, D. Young, et al., ed., Acad. Press, 1990.Google Scholar
  8. 8.
    S. Eisenstat,On the rate of convergence of the Bergman-Vekua method for the numerical solution of elliptic boundary value problems, SIAM J. Numer. Anal., 11 (1974), pp. 654–680.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    G. Fairweather and L. Johnston,The method of fundamental solutions for problems in potential theory, in Treatment of Integral Equations by Numerical Methods, C. Baker and G. Miller, ed., Acad. Press, 1982.Google Scholar
  10. 10.
    K. Gallivan, W. Jalby andU. Meier,The use of BLAS3 in linear algebra on a parallel processor with hierarchical memory, SIAM J. Sci. Stat. Comp., 11 (1987), pp. 1079–1084.Google Scholar
  11. 11.
    R. Glowinski, et al, ed.,Domain decomposition methods for partial differential equations, SIAM, 1991.Google Scholar
  12. 12.
    G. Golub andC. Van Loan,Matrix Computations, 2nd ed., The Johns Hopkins University Press, Baltimore, MD, 1989.MATHGoogle Scholar
  13. 13.
    P. Henrici,Applied and Computational Complex Analysis, V.3, Wiley, 1986.Google Scholar
  14. 14.
    R. Hockney,Rapid elliptic solvers, in Numerical Methods in Applied Fluid Dynamics, 1980.Google Scholar
  15. 15.
    Z. Li andR. Mathon,Error and stability analysis of boundary methods for elliptic problems with interfaces, Math. Comp., 54 (1990), pp. 41–61.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Z. Li, R. Mathon andP. Sermer,Boundary methods for solving elliptic problems with singularities and interfaces, SIAM J. Numer. Anal., 24 (1987), pp. 487–498.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    R. Mathon andR. Johnston,The approximate solution of elliptic boundary value problems by fundamental solutions, SIAM J. Numer. Anal., 14 (1977), pp. 638–650.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    A. Mayo,The fast solution of Poisson and biharmonic equations on irregular regions, SIAM J. Numer. Anal., 21 (1984), pp. 285–299.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    P. Swarztrauber andR. Sweet,Efficient FORTRANsubprograms for the solution of elliptic partial differential equations, ACM Trans. Math. Software, 5 (1979), pp. 352–364.MATHCrossRefGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2001

Authors and Affiliations

  1. 1.Information and Communications Engineering DivisionHallym UniversityChunchonKorea

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