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Numerical conformal mapping of multiply connected domains to regions with circular boundaries

Wei Luo Junfei Dai Xianfeng Gu Shing-Tung Yau

Computational Geometry mathscidoc:1912.43646

Journal of computational and applied mathematics, 233, (11), 2940-2947, 2010.4
We propose a method to map a multiply connected bounded planar region conformally to a bounded region with circular boundaries. The norm of the derivative of such a conformal map satisfies the Laplace equation with a nonlinear Neumann type boundary condition. We analyze the singular behavior at corners of the boundary and separate the major singular part. The remaining smooth part solves a variational problem which is easy to discretize. We use a finite element method and a gradient descent method to find an approximate solution. The conformal map is then constructed from this norm function. We tested our algorithm on a polygonal region and a curvilinear smooth region.
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@inproceedings{wei2010numerical,
  title={Numerical conformal mapping of multiply connected domains to regions with circular boundaries},
  author={Wei Luo, Junfei Dai, Xianfeng Gu, and Shing-Tung Yau},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224204718462812210},
  booktitle={Journal of computational and applied mathematics},
  volume={233},
  number={11},
  pages={2940-2947},
  year={2010},
}
Wei Luo, Junfei Dai, Xianfeng Gu, and Shing-Tung Yau. Numerical conformal mapping of multiply connected domains to regions with circular boundaries. 2010. Vol. 233. In Journal of computational and applied mathematics. pp.2940-2947. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224204718462812210.
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