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

Jiayao Zhang; Yibin Lu; Yue Shan; Yingzi Wang; Fuming Lai; Jiayao Zhang; Yibin Lu; Yue Shan; Yingzi Wang; Fuming Lai

doi:10.3934/era.2025341

Electronic Research Archive

2025, Volume 33, Issue 12: 7717-7735. doi: 10.3934/era.2025341

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Research article

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

1.
Faculty of Science, Kunming University of Science and Technology, Kunming 650500, China
2.
Faculty of Information Engineering and Automation, Kunming University of Science and Technology, Kunming 650500, China
3.
School of Sustainable Energy and Materials Science, Jinhua Advanced Research Institute, Jinhua 321013, China

Received: 16 October 2025 Revised: 02 December 2025 Accepted: 08 December 2025 Published: 22 December 2025

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.
Citation: Jiayao Zhang, Yibin Lu, Yue Shan, Yingzi Wang, Fuming Lai. A new result for numerical conformal mapping of bounded multiply connected domains[J]. Electronic Research Archive, 2025, 33(12): 7717-7735. doi: 10.3934/era.2025341

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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.

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