• Ronny Ramlau, Lothar Reichel (Hg.)

ETNA - Electronic Transactions on Numerical Analysis

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Electronic Transactions on Numerical Analysis (ETNA) is an electronic journal for the publication of significant new developments in numerical analysis and scientific computing. Papers of the highest quality that deal with the analysis of algorithms for the solution of continuous models and numerical linear algebra are appropriate for ETNA, as are papers of similar quality that discuss implementation and performance of such algorithms. New algorithms for current or new computer architectures are appropriate provided that they are numerically sound. However, the focus of the publication should be on the algorithm rather than on the architecture. The journal is published by the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM). Reviews of all ETNA papers appear in Mathematical Reviews and Zentralblatt für Mathematik. Reference information for ETNA papers also appears in the expanded Science Citation Index. ETNA is registered with the Library of Congress and has ISSN 1068-9613.

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Verlag der Österreichischen Akademie der Wissenschaften
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ETNA - Electronic Transactions on Numerical Analysis



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Verlag der Österreichischen Akademie der Wissenschaften
Austrian Academy of Sciences Press
A-1011 Wien, Dr. Ignaz Seipel-Platz 2,
Tel. +43-1-515 81/DW 3420, Fax +43-1-515 81/DW 3400
https://verlag.oeaw.ac.at, e-mail: bestellung.verlag@oeaw.ac.at
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Efficient numerical conformal mappings on multiply connected Riemann surface

    Harri Hakula, Antti Rasila, Yufan Zheng

ETNA - Electronic Transactions on Numerical Analysis, pp. 533-552, 2026/10/20

doi: 10.1553/etna_vol65s533

doi: 10.1553/etna_vol65s533


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doi:10.1553/etna_vol65s533



doi:10.1553/etna_vol65s533

Abstract

The conjugate function method is an algorithm for the numerical computation of conformal mappings for simply and multiply connected domains on surfaces. In this paper, the conjugate function method, previously used for simply connected domains, is generalized and refined to achieve the same level of accuracy on multiply connected planar domains and Riemann surfaces. The main challenge is the accurate and efficient construction of boundary values for the conjugate problem on multiply connected domains. The method relies on high-order finite element methods, which allow for highly accurate computations of mappings on surfaces, including domains of complex boundary geometry containing strong singularities and cusps. We also derive the reciprocal error estimate for the multiply connected case. The efficacy of the proposed method is illustrated through an extensive set of numerical experiments with error estimates.

Keywords: Laplace–Beltrami, numerical conformal mappings, conformal modulus, Riemann surfaces