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Numerical Computation of the Conformal Map onto Lemniscatic Domains

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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.

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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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Correspondence to Olivier Sète.

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Communicated by Darren Crowdy.

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Nasser, M.M.S., Liesen, J. & Sète, O. Numerical Computation of the Conformal Map onto Lemniscatic Domains. Comput. Methods Funct. Theory 16, 609–635 (2016). https://doi.org/10.1007/s40315-016-0159-x

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