A new result for numerical conformal mapping of bounded multiply connected domains

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

This paper proposes a novel numerical scheme based on an improved charge simulation method for computing conformal mappings of bounded multiply connected domains. The core of the method reformulates the mapping problem into a constraint system constructed via charge simulation. This system is then solved efficiently using the symmetric successive over-relaxation incomplete Cholesky conjugate gradient method, which is particularly well-suited for handling the ill-conditioned systems inherent to such problems. Numerical experiments show that our method achieves significantly higher accuracy and improved convergence rates compared to the conventional Gauss-Seidel iteration. The results confirm the robustness and practical potential of the proposed framework, establishing it as an efficient and reliable tool for computing conformal mappings of domains with high connectivity and complex geometry.

Keywords

charge simulation method / conformal mapping / bounded multiply connected domains / symmetric successive over-relaxation incomplete Cholesky conjugate gradient method / ill-conditioned systems

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] D. Crowdy, Conformal slit maps in applied mathematics, Anziam J., 53 (2012), 171-189. https://doi.org/10.1017/S1446181112000119

[2] K. He, J. Chang, D. Pang, B. Sun, Z. Yin, D. Li, Iterative algorithm for the conformal mapping function from the exterior of a roadway to the interior of a unit circle, Arch. Appl. Mech., 92 (2022), 971-991. https://doi.org/10.1007/s00419-021-02087-w

[3] D. Ntalampekos, Rigidity and continuous extension for conformal maps of circle domains, Trans. Am. Math. Soc., 376 (2023), 5221-5239. https://doi.org/10.1090/tran/8923

[4] T. Bergamaschi, W. I. Jay, P. R. Oare, Hadronic structure, conformal maps, and analytic continuation, Phys. Rev. D, 108 (2023), 074516, https://doi.org/10.1103/PhysRevD.108.074516

[5] G. T. Symm, An integral equation method in conformal mapping, Numer. Math., 9 (1966), 250-258. https://doi.org/10.1007/bf02162088

[6] G. T. Symm, Conformal mapping of doubly-connected domains, Numer. Math., 13 (1969), 448-457. https://doi.org/10.1007/bf02163272

[7] K. Amano, A charge simulation method for the numerical conformal mapping of interior, exterior and doubly-connected domains, J. Comput. Appl. Math., 53 (1994), 353-370. https://doi.org/10.1016/0377-0427(94)90063-9

[8] D. Okano, H. Ogata, K. Amano, M. Sugihara, Numerical conformal mappings of bounded multiply connected domains by the charge simulation method, J. Comput. Appl. Math., 159 (2003), 109-117. https://doi.org/10.1016/s0377-0427(03)00572-7

[9] K. Amano, D. Okano, A circular and radial slit mapping of unbounded multiply connected domains, JSIAM Lett., 2 (2010), 53-56. https://doi.org/10.14495/jsiaml.2.53

[10] K. Amano, D. Okano, H. Ogata, M. Sugihara, Numerical conformal mappings onto the linear slit domain, Jpn. J. Ind. Appl. Math., 29 (2012), 165-186. https://doi.org/10.1007/s13160-012-0058-0

[11] M. Nasser, O. Rainio, A. Rasila, M. Vuorinen, T. Wallace, H. Yu, et al., Polycircular domains, numerical conformal mappings, and moduli of quadrilaterals, Adv. Comput. Math., 48 (2022), 58. https://doi.org/10.1007/s10444-022-09975-x

[12] M. M. S. Nasser, A boundary integral equation for conformal mapping of bounded multiply connected regions, Comput. Methods Funct. Theory, 9 (2009), 127-143. https://doi.org/10.1007/bf03321718

[13] D. Crowdy, The Schwarz-Christoffel mapping to bounded multiply connected polygonal domains, Proc. R. Soc. A: Math. Phys. Eng. Sci., 461 (2005), 2653-2678. https://doi.org/10.1098/rspa.2005.1480

[14] D. Crowdy, J. Marshall, Conformal mappings between canonical multiply connected domains, Comput. Methods Funct. Theory, 6 (2006), 59-76. https://doi.org/10.1007/BF03321118

[15] X. Gu, Y. Wang, T. F. Chan, P. M. Thompson, S. Yau, Genus zero surface conformal mapping and its application to brain surface mapping, IEEE Trans. Med. Imaging, 23 (2004), 949-958. https://doi.org/10.1109/TMI.2004.831226

[16] X. Gu, F. Luo, S. Yau, Computational conformal geometry behind modern technologies, Not. Am. Math. Soc., 67 (2020), 1509-1525. https://doi.org/10.1090/noti2164

[17] H. Hakula, T. Quach, A. Rasila, The conjugate function method and conformal mappings in multiply connected domains, SIAM J. Sci. Comput., 41 (2019), 1753-1776. https://doi.org/10.1137/17M1124164

[18] H. Hakula, A. Rasila, Laplace-beltrami equation and numerical conformal mappings on surfaces, SIAM J. Sci. Comput., 47 (2025), 325-342. https://doi.org/10.1137/24M1656840

[19] M. M. S. Nasser, Numerical conformal mapping of multiply connected regions onto the fifth category of koebe’s canonical slit regions, J. Math. Anal. Appl., 398 (2013), 729-743. https://doi.org/10.1016/j.jmaa.2012.09.020

[20] M. M. S. Nasser, 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 (2011), 47-56. https://doi.org/10.1016/j.jmaa.2011.04.030

[21] T. Wang, D. Huang, G. Ma, Z. Meng, Y. Li, Improved preconditioned conjugate gradient algorithm and application in 3D inversion of gravity-gradiometry data, Appl. Geophys., 14 (2017), 301-313. https://doi.org/10.1007/s11770-017-0625-x

[22] C. Vuik, Krylov subspace solvers and preconditioners, in ESAIM: Proceedings and Surveys, 63 (2018), 1-43. https://doi.org/10.1051/proc/201863001

[23] O. Axelsson, A generalized SSOR method, BIT Numer. Math., 12 (1972), 443-467. https://doi.org/10.1007/bf01932955

[24] H. Hakula, A. Rasila, Y. Zheng, The conjugate function method for surfaces with elaborate topological types, preprint, arXiv:2509.01978.