Journal of Scientific Computing

, Volume 78, Issue 1, pp 582–606 | Cite as

Numerical Computing of Preimage Domains for Bounded Multiply Connected Slit Domains

  • Mohamed M. S. NasserEmail author


In this paper, for a given bounded multiply connected slit domain \(\varOmega \), we present an iterative numerical method for computing a conformally equivalent multiply connected domain G bounded by smooth Jordan curves and the conformal mapping \(w=\varPhi (z)\) from G onto \(\varOmega \). Each iteration of the proposed iterative method requires solving the boundary integral equation with the generalized Neumann kernel. We consider two cases of bounded slit domains, namely the unit disk with radial slit domain and an annulus with radial slit domain. Numerical examples are presented to illustrate that the proposed iterative method converges even for highly connected slit domains.


Numerical conformal mapping Generalized Neumann kernel Multiply connected domains Canonical slit domains 

Mathematics Subject Classification

30C30 45B05 65E05 



The author is grateful to an anonymous referee for his valuable comments and suggestions which improved the results and the presentation of this paper. Further, the author thanks Prof. Leslie Greengard and Dr. Zydrunas Gimbutas for making the MATLAB toolbox FMMLIB2D [17] publicly available.


  1. 1.
    Amano, K.: A charge simulation method for numerical conformal mapping onto circular and radial slit domains. SIAM J. Sci. Comput. 19, 1169–1187 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aoyama, N., Sakajo, T., Tanaka, H.: A computational theory for spiral point vortices in multiply connected domains with slit boundaries. Jpn. J. Ind. Appl. Math. 30, 485–509 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Atkinson, K.: The Numerical Solution of Integral Equations of the Second Kind. Cambridge University Press, Cambridge (1997)CrossRefzbMATHGoogle Scholar
  4. 4.
    Benchama, N., DeLillo, T., Hrycak, T., Wang, L.: A simplified Fornberg-like method for the conformal mapping of multiply connected regions-comparisons and crowding. J. Comput. Appl. Math. 209, 1–21 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bourchtein, L.: Conformal mappings of multiply connected domains onto canonical domains using the Green and Neumann functions. Complex Var. Elliptic Equ. 58(6), 821–836 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Crowdy, D.: Analytical solutions for uniform potential flow past multiple cylinders. Eur. J. Mech. B Fluids 25, 459–470 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Crowdy, D.: Calculating the lift on a finite stack of cylindrical aerofoils. Proc. R. Soc. A 462, 1387–1407 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Crowdy, D.: Explicit solution for the potential flow due to an assembly of stirrers in an inviscid fluid. J. Eng. Math. 62, 333–344 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Crowdy, D.: A new calculus for two-dimensional vortex dynamics. Theor. Comput. Fluid Dyn. 24, 9–24 (2010)CrossRefzbMATHGoogle Scholar
  10. 10.
    Crowdy, D.: Conformal slit maps in applied mathematics. ANZIAM J. 53, 171–189 (2012)zbMATHGoogle Scholar
  11. 11.
    Crowdy, D., Kropf, E.H., Green, C.C., Nasser, M.: The Schottky–Klein prime function: a theoretical and computational tool for applications. IMA J. Appl. Math. 81, 589–628 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Crowdy, D., Marshall, J.: Conformal mappings between canonical multiply connected domains. Comput. Methods Funct. Theory 6(1), 59–76 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    DeLillo, T.: The accuracy of numerical conformal mapping methods: a survey of examples and results. SIAM J. Numer. Anal. 31, 788–812 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    DeLillo, T., Driscoll, T., Elcrat, A., Pfaltzgraff, J.: Radial and circular slit maps of unbounded multiply connected circle domains. Proc. R. Soc. A 464(2095), 1719–1737 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    DeLillo, T., Elcrat, A.: A Fornberg-like conformal mapping method for slender regions. J. Comput. Appl. Math. 46, 49–64 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Goluzin, G.: Geometric Theory of Functions of a Complex Variable. Amer. Math. Soc., Rhode Island (1969)CrossRefzbMATHGoogle Scholar
  17. 17.
    Greengard, L., Gimbutas, Z.: FMMLIB2D: A MATLAB toolbox for fast multipole method in two dimensions. Version 1.2 (2012). Accessed 1 Jan 2018
  18. 18.
    Gutknecht, M.: Numerical experiments on solving theodorsen’s integral equation for conformal maps with the fast fourier transform and various nonlinear iterative methods. SIAM J. Sci. Stat. Comput. 4(1), 1–30 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Henrici, P.: Applied and Computational Complex Analysis, vol. 3. Wiley, New York (1986)zbMATHGoogle Scholar
  20. 20.
    Kerzman, N., Trummer, M.: Numerical conformal mapping via the Szegö kernel. J. Comput. Appl. Math. 14, 111–123 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Koebe, P.: Über die konforme Abbildung mehrfach-zusammenhängender Bereiche. Jahresber. Deut. Math. Ver. 19, 339–348 (1910)zbMATHGoogle Scholar
  22. 22.
    Koebe, P.: Abhandlungen zur theorie der konformen abbildung, iv. abbildung mehrfach zusammenhängender schlichter bereiche auf schlitzbe-reiche. Acta Math. 41, 305–344 (1918)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kress, R.: Linear Integral Equations, 3rd edn. Springer, New York (2014)CrossRefzbMATHGoogle Scholar
  24. 24.
  25. 25.
    Nasser, M.M.S.: A boundary integral equation for conformal mapping of bounded multiply connected regions. Comput. Methods Funct. Theory 9, 127–143 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Nasser, M.M.S.: Numerical conformal mapping via a boundary integral equation with the generalized Neumann kernel. SIAM J. Sci. Comput. 31(3), 1695–1715 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Nasser, M.M.S.: Numerical conformal mapping of multiply connected regions onto the second, third and fourth categories of Koebe’s canonical slit domains. J. Math. Anal. Appl. 382, 47–56 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Nasser, M.M.S.: Numerical conformal mapping of multiply connected regions onto the fifth category of Koebe’s canonical slit regions. J. Math. Anal. Appl. 398, 729–743 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Nasser, M.M.S.: Fast computation of the circular map. Comput. Methods Funct. Theory 15(2), 187–223 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Nasser, M.M.S.: Fast solution of boundary integral equations with the generalized Neumann kernel. Electron. Trans. Numer. Anal. 44, 189–229 (2015)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Nasser, M.M.S.: CircularMap: A numerical implementation of the circular map in MATLAB (2017). Accessed 1 Jan 2018
  32. 32.
    Nasser, M.M.S., Al-Shihri, F.: A fast boundary integral equation method for conformal mapping of multiply connected regions. SIAM J. Sci. Comput. 35(3), A1736–A1760 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Nasser, M.M.S., Green, C.C.: A fast numerical method for ideal fluid flow in domains with multiple stirrers. Nonlinearity 31, 815–837 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Okano, D., Ogata, H., Amano, K.: A method of numerical conformal mapping of curved slit domains by the charge simulation method. J. Comput. Appl. Math. 152, 441–450 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Pommerenke, C.: On the logarithmic capacity and conformal mapping. Duke Math. J. 35, 321–325 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Razali, M., Nashed, M., Murid, A.: Numerical conformal mapping via the Bergman kernel. J. Comput. Appl. Math. 82, 335–350 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Sangawi, A., Murid, A.H.M., Nasser, M.M.S.: Radial slit maps of bounded multiply connected regions. J. Sci. Comput. 55, 309–326 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Schiffer, M.: Some recent developments in the theory of conformal mapping. In: Appendix to: R. Courant, Dirichlet’s principle, conformal mapping and minimal surfaces. Interscience, New York (1950)Google Scholar
  39. 39.
    Trummer, M.: An efficient implementation of a conformal mapping method based on the Szegö kernel. SIAM J. Numer. Anal. 23(4), 853–872 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Wegmann, R.: Crowding for analytic functions with elongated range. Constr. Approx. 10, 179–186 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Wegmann, R.: Methods for numerical conformal mapping. In: Kühnau, R. (ed.) Handbook of Complex Analysis: Geometric Function Theory, vol. 2, pp. 351–477. Elsevier, New York (2005)CrossRefGoogle Scholar
  42. 42.
    Wegmann, R., Nasser, M.M.S.: The Riemann–Hilbert problem and the generalized Neumann kernel on multiply connected regions. J. Comput. Appl. Math. 214, 36–57 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Yunus, A., Murid, A.H.M., Nasser, M.M.S.: Numerical conformal mapping and its inverse of unbounded multiply connected regions onto logarithmic spiral slit regions and rectilinear slit regions. Proc. R. Soc. A. 470(2162), 514 (2014)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Statistics and PhysicsQatar UniversityDohaQatar

Personalised recommendations