Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Numerical Conformal Mapping with Rational Functions

  • Published:
Computational Methods and Function Theory Aims and scope Submit manuscript

Abstract

New algorithms are presented for numerical conformal mapping based on rational approximations and the solution of Dirichlet problems by least-squares fitting on the boundary. The methods are targeted at regions with corners, where the Dirichlet problem is solved by the “lightning Laplace solver” with poles exponentially clustered near each singularity. For polygons and circular polygons, further simplifications are possible.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

Notes

  1. Gaier: “Since the configuration of the maximal system of points on the boundary is unknown (except for circles and ellipses), this method is mainly of theoretical interest.” Curtis: “The success of this method is highly sensitive to the correct placement of the points.” Collatz: “The choice of collocation points is a matter of some uncertainty.”

References

  1. Adcock, B., Huybrechs, D.: Frames and numerical approximation. SIAM Rev. 61, 443–473 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  2. Amano, K.: A charge simulation method for the numerical conformal mapping of interior, exterior and doubly-connected domains. J. Comput. Appl. Math. 53, 353–370 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  3. Amano, K., Okano, D., Ogata, H., Sugihara, M.: Numerical conformal mappings onto the linear slit domain. Jpn. J. Ind. Appl. Math. 29, 165–186 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Axler, S.: Harmonic functions from a complex anlaysis viewpoint. Am. Math. Mon. 39, 246–258 (1986)

    Article  MATH  Google Scholar 

  5. Badreddine, M., DeLillo, T.K., Sahraei, S.: A comparison of some numerical conformal mapping methods for simply and multiply connected domains. Discret. Contin. Dyn. Syst. Ser. B 24, 55–82 (2019)

    MathSciNet  MATH  Google Scholar 

  6. Barnett, A.H., Betcke, T.: Stability and convergence of the method of fundamental solutions for Helmholtz problems on analytic domains. J. Comput. Phys. 227, 7003–7026 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Beckermann, B., Townsend, A.: Bounds on the sngular valuces of matrices with displacement structure. SIAM Rev. 61, 319–344 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bishop, C.J.: Conformal welding and Koebe’s theorem. Ann. Math. 166, 613–656 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bjørstad, P., Grosse, E.: Conformal mapping of cirulcar arc polygons. SIAM J. Sci. Comput. 8, 19–32 (1987)

    Article  MATH  Google Scholar 

  10. Brubeck, P.D., Nakatsukasa, Y., Trefethen, L.N.: Vandermonde with Arnoldi (2019). arXiv:1911.09988 (submitted to SIAM Review)

  11. Collatz, L.: The numerical treatment of differential equations, 3rd edn. Springer, New York (1960)

    Book  MATH  Google Scholar 

  12. Computational Methods and Function Theory, special issue on numerical conformal mapping, vol. 11, no. 2, pp. 375–787 (2012)

  13. Crowdy, D.: The Schwarz-Christoffel mapping to bounded multiply connected polygonal domains. Proc. R. Soc. A 461, 2653–2678 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Curtiss, J.H.: Solutions of the Dirichlet problem in the plane by approximation with Faber polynomials. SIAM J. Numer. Anal. 3, 204–228 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  15. Däppen, H.D.: Die Schwarz-Christoffel-Abbildung für zweifach zusammenhängende Gebiete mit Anwendungen. Diss, ETH Zurich (1988)

    MATH  Google Scholar 

  16. DeLillo, T.K., Elcrat, A.R., Kropf, E.H., Pfaltzgraff, J.A.: Efficient calculation of Schwarz-Christoffel transformations for multiply connected domains using Laurent series. Comput. Methods Funct. Theory 13, 307–336 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. DeLillo, T.K., Elcrat, A.R., Pfaltzgraff, J.A.: Schwarz-Christoffel mapping of multiply connected domains. J. Anal. Math. 94, 17–47 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. Driscoll, T.A.: Algorithm 756: A MATLAB toolbox for Schwarz–Christoffel mapping. ACM Trans. Math. Softw., vol. 22, pp. 168–186 (1996) (see also www.math.udel.edu/~driscoll/SC/)

  19. Driscoll, T.A., Hale, N., Trefethen, L.N. (eds.): Chebfun User’s Guide. Pafnuty Publications, Oxford (2014). www.chebfun.org

  20. Driscoll, T.A., Trefethen, L.N.: Schwarz-Christoffel Mapping. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Gaier, D.: Konstruktive Methoden der konformen Abbildung. Springer, New York (1964)

    Book  MATH  Google Scholar 

  23. Gaier, D.: Lectures on complex approximation. Birkhäuser, Basel (1987)

    Book  MATH  Google Scholar 

  24. Gopal, A., Trefethen, L.N.: Representation of conformal maps by rational functions. Numer. Math. 142, 359–382 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gopal, A., Trefethen, L.N.: Solving Laplace problems with corner singularities via rational functions. SIAM J. Numer. Anal. 57, 2074–2094 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  26. Henrici, P.: Applied and Computational Complex Analysis. Wiley, New York (1974)

    MATH  Google Scholar 

  27. Howell, L.H.: Numerical conformal mapping of circular arc polygons. J. Comput. Appl. Math. 46, 7–28 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  28. Hu, C.: Algorithm 785: A software package for computing Schwarz-Christoffel conformal transformation for doubly connected polygonal regions. ACM Trans. Math. Softw. 24, 317–333 (1998)

    Article  MATH  Google Scholar 

  29. Kerzman, N., Stein, E.M.: The Cauchy kernel, the Szegő kernel, and the Riemann mapping function. Math. Ann. 236(10978), 85–93 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kerzman, N., Trummer, M.R.: Numerical conformal mapping via the Szegő kernel. J. Comput. Appl. Math. 14, 111–123 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  31. Lehman, R.S.: Development of the mapping function at an analytic corner. Pac. J. Math. 7, 1437–1449 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lehman, R.S.: Developments at an analytic corner of solutions of elliptic partial differential equations. J. Math. Mech. 8, 727–760 (1959)

    MathSciNet  MATH  Google Scholar 

  33. Nakatsukasa, Y., Sète, O., Trefethen, L.N.: The AAA algorithm for rational approximation. SIAM J. Sci. Comput. 40, A1494–A1522 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  34. Nakatsukasa, Y., Trefethen, L.N.: An algorithm for real and complex rational minimax approximation. SIAM J. Sci. Comput. (2019) (submitted)

  35. Nasser, M.M.S.: PlgCirMap: A MATLAB toolbox for computing the conformal mapping from polygonal multiply connected domains onto circular domains SoftwareX,11, 100464 (2020).

  36. Newman, D.J.: Rational approximation to \(|x|\). Mich. Math. J. 11, 11–14 (1964)

    MATH  Google Scholar 

  37. Schiffer, M.: Some recent developments in the theory of conformal mapping, appendix to R. Dirichlet’s Principle, Conformal Mapping, and Minimal Surfaces, Interscience, Courant (1950)

    Google Scholar 

  38. Stahl, H.: Spurious poles in Padé approximation. J. Comput. Appl. Math. 99, 511–527 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  39. Townsend, A., Wilber, H.: On the singular values of matrices with large displacement rank. Linear Algebra Appl. 548, 19–41 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  40. Trefethen, L.N. (ed.): Numerical conformal mapping. North-Holland (1986)

  41. Trefethen, L.N.: Series solution of Laplace problems. ANZIAM J. 60, 1–26 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  42. Trefethen, L.N.: Conformal mapping of L-shaped regions. (2019). Chebfun example at www.chebfun.org(October 2019)

  43. Trefethen, L.N.: Lightning Laplace and conformal mapping codes laplace.m and confmap.m, people.maths.ox.ac.uk/trefethen/laplace (2019)

  44. Walsh, J.L.: Interpolation and approximation by rational functions in the complex domain. American Mathematical Society, RI (1935)

  45. Wegmann, R.: Methods for numerical conformal mapping. In: Kühnau, R. (ed.) Handbook of complex analysis: geometric function theory, vol. 2, pp. 351–477. Elsevier, Amsterdam (2020)

    Chapter  Google Scholar 

Download references

Acknowledgements

I have benefited from helpful comments of Toby Driscoll, Abi Gopal, and Yuji Nakatsukasa.

Author information

Authors and Affiliations

  1. Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK

    Lloyd N. Trefethen

Authors
  1. Lloyd N. Trefethen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lloyd N. Trefethen.

Additional information

Communicated by In Memoriam Stephan Ruscheweyh 1944–2019. Elias Wegert.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Trefethen, L.N. Numerical Conformal Mapping with Rational Functions. Comput. Methods Funct. Theory 20, 369–387 (2020). https://doi.org/10.1007/s40315-020-00325-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40315-020-00325-w

Keywords

Mathematics Subject Classification

Search

Navigation