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Computational Methods and Function Theory

, Volume 16, Issue 4, pp 609–635 | Cite as

Numerical Computation of the Conformal Map onto Lemniscatic Domains

  • Mohamed M. S. Nasser
  • Jörg Liesen
  • Olivier SèteEmail author
Article

Abstract

We present a numerical method for the computation of the conformal map from unbounded multiply-connected domains onto lemniscatic domains. For \(\ell \)-times connected domains, the method requires solving \(\ell \) boundary integral equations with the Neumann kernel. This can be done in \(O(\ell ^2 n \log n)\) operations, where n is the number of nodes in the discretization of each boundary component of the multiply-connected domain. As demonstrated by numerical examples, the method works for domains with close-to-touching boundaries, non-convex boundaries, piecewise smooth boundaries, and for domains of high connectivity.

Keywords

Numerical conformal mapping Multiply-connected domains Lemniscatic domains Boundary integral equations  Neumann kernel 

Mathematics Subject Classification

30C30 45B05 65E05 

Notes

Acknowledgments

We thank Robert Luce for helpful discussions on Cauchy matrices and the solution of the systems (5.9)–(5.10). We also thank Elias Wegert for suggesting our collaboration. We further thank the anonymous referees for helpful comments.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Mohamed M. S. Nasser
    • 1
  • Jörg Liesen
    • 2
  • Olivier Sète
    • 2
    Email author
  1. 1.Department of Mathematics, Statistics and Physics, College of Arts and SciencesQatar UniversityDohaQatar
  2. 2.TU Berlin, MA 4-5BerlinGermany

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