Volume 43 Issue 9
Sep.  2022
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WU Kang, LÜ Yibin, SHI Yunlong, WANG Yingzi. The GMRES(m) Method for Numerical Conformal Mapping of Bounded Multi-Connected Domains[J]. Applied Mathematics and Mechanics, 2022, 43(9): 1026-1033. doi: 10.21656/1000-0887.420305
Citation: WU Kang, LÜ Yibin, SHI Yunlong, WANG Yingzi. The GMRES(m) Method for Numerical Conformal Mapping of Bounded Multi-Connected Domains[J]. Applied Mathematics and Mechanics, 2022, 43(9): 1026-1033. doi: 10.21656/1000-0887.420305

The GMRES(m) Method for Numerical Conformal Mapping of Bounded Multi-Connected Domains

doi: 10.21656/1000-0887.420305
  • Received Date: 2021-10-11
  • Rev Recd Date: 2022-03-06
  • Available Online: 2022-09-07
  • Publish Date: 2022-09-30
  • It is difficult to solve conformal mapping functions for complex multi-connected domains. In order to overcome this difficulty, the problem of solving conformal mapping functions was transformed into using the charge simulation method to solve a pair of conjugate harmonic functions in the problem domain. The conjugate harmonic functions should satisfy given boundary conditions, which construct a system of linear equations. Then the simulation charges can be computed by means of the GMRES(m) (the generalized minimal residual method) algorithm to solve the linear systems. The approximate conformal mapping functions were constructed accurately to map the bounded multi-connected domain onto 3 unbounded canonical slit domains. Numerical results show that the presented method is effective.

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  • [1]
    LYU Y Y, JIANG J, WANG Y L, et al. Superconducting diode effect via conformal-mapped nanoholes[J]. Nature Communications, 2021, 12: 2703. doi: 10.1038/s41467-021-23077-0
    [2]
    朱满座. 数值保角变换及其在电磁理论中的应用[D]. 博士学位论文. 西安: 西安电子科技大学, 2008.

    ZHU Manzuo. Numerical conformal mapping and its applications in the electromagnetic theory[D]. PhD Thesis. Xi’an: Xidian University, 2008. (in Chinese)
    [3]
    曹伟杰. 保形变换理论及其应用[M]. 上海: 上海科学技术文献出版社, 1988.

    CAO Weijie. Conformal Mapping Theory and Its Application[M]. Shanghai: Shanghai Scientific and Technical Literature Press, 1988. (in Chinese)
    [4]
    CONSTANTIN A, STRAUSS W, VARVARUCA E. Global bifurcation of steady gravity water waves with critical layers[J]. Acta Mathematica, 2016, 217: 195-262. doi: 10.1007/s11511-017-0144-x
    [5]
    SUNAO M, WOOYOUNG C. A numerical study on parasitic capillary waves using unsteady conformal mapping[J]. Journal of Computational Physics, 2017, 328: 234-257. doi: 10.1016/j.jcp.2016.10.015
    [6]
    郑志强. 单位圆到任意曲线保角变换的近似计算方法[J]. 应用数学和力学, 1992, 13(5): 449-457

    ZHENG Zhiqiang. An approximate method on the conformal mapping from a unit circle to an arbitrary curve[J]. Applied Mathematics and Mechanics, 1992, 13(5): 449-457.(in Chinese)
    [7]
    SANGAWI A W K, MURID A H M, NASSER M M S. Annulus with circular slit map of bounded multiply connected regions via integral equation method[J]. Bulletin of the Malaysian Mathematical Society, 2012, 4(4): 945-959.
    [8]
    KOKKINOS C A, PAPAMICHAEL N, SIDERIDIS A B. An orthonormalization method for the approximate conformal mapping of multiply-connected domains[J]. IMA Journal of Numerical Analysis, 1990, 10(3): 343-359. doi: 10.1093/imanum/10.3.343
    [9]
    TREFETHEN L N. Numerical conformal mapping with rational functions[J]. Computational Methods and Function Theory, 2020, 20: 369-387. doi: 10.1007/s40315-020-00325-w
    [10]
    LO W L, WU N J, CHEN C S, et al. Exact boundary derivative formulation for numerical conformal mapping method[J]. Mathematical Problems in Engineering, 2016, 2: 1-18.
    [11]
    AMANO K. Numerical conformal mapping of exterior domains based on the charge simulation method[J]. Information Processing Society of Japan Journal, 1988, 29(1): 62-72.
    [12]
    AMANO K. A charge simulation method for the numerical conformal mapping of interior, exterior and doubly-connected domains[J]. Journal of Computational and Applied Mathematics, 1994, 53: 353-370. doi: 10.1016/0377-0427(94)90063-9
    [13]
    NEHARI Z. Conformal Mapping[M]. New York: Dover Publications, 1952.
    [14]
    OKANO D, OGATA H, AMANO K, et al. Numerical conformal mappings of bounded multiply connected domains by the charge simulation method[J]. Journal of Computational and Applied Mathematics, 2003, 159(1): 109-117. doi: 10.1016/S0377-0427(03)00572-7
    [15]
    吕毅斌, 赖富明, 王樱子, 等. 基于GMRES(m)法的双连通区域数值保角变换的计算法[J]. 数学杂志, 2016, 36(5): 1028-1034 doi: 10.3969/j.issn.0255-7797.2016.05.017

    LÜ Yibin, LAI Fuming, WANG Yingzi, et al. The GMRES(m) method for numerical conformal mapping of doubly-connected domain[J]. Journal of Mathematics, 2016, 36(5): 1028-1034.(in Chinese) doi: 10.3969/j.issn.0255-7797.2016.05.017
    [16]
    MUROTA K. Comparison of conventional and “invariant” schemes of fundamental solutions method for annular domains[J]. Japan Journal of Industrial and Applied Mathematics, 1995, 12(1): 61-85. doi: 10.1007/BF03167382
    [17]
    SAAD Y, SCHULTZ M H. GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems[J]. SIAM Journal on Scientific and Statistical Computing, 1986, 7(3): 856-869. doi: 10.1137/0907058
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