A Fast Algorithm for the Numerical Evaluation of Conformal Mappings

Siam Journal on Scientific and Statistical Computing Pub Date : 1989-05-01 DOI:10.1137/0910031

S. O'Donnell, V. Rokhlin

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引用次数: 61

Abstract

An algorithm is presented for the construction of conformal mappings from arbitrary simply connected regions in the complex plane onto the unit disk. The algorithm is based on a combination of the Kerzman–Stein integral equation (see [Math. Anal, 236 (1978), pp. 85–93]) and the Fast Multipole Method for the evaluation of Cauchy-type integrals (see [V. Rokhlin, J. Comput. Phys., 60 (1985), pp. 187–207], [L. Greengard and V. Rokhlin, J. Comput. Phys., 73 (1987), pp. 325–348], [J. Carrier, L. Greengard, and V. Rokhlin, SIAM J. Sci. Statist. Comput., 9 (1988), pp. 669–686], [L. F. Greengard, Ph.D. thesis, Department of Computer Science, Yale University, New Haven, CT, 1987]). Previously published methods for the construction of conformal mappings via the Kerzman–Stein equation have an asymptotic CPU time estimate of the order $O(n^2 )$, where n is the number of nodes in the discretization of the boundary of the region being mapped. The method presented here has an estimate of the order $O(n)$, making it an approach of choice in many situations. The performance of the algorithm is illustrated by several numerical examples.

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保角映射数值求值的快速算法

给出了复平面上任意单连通区域到单位圆盘的保角映射的构造算法。该算法基于Kerzman-Stein积分方程(参见[数学])的组合。数学学报，236 (1978)，pp. 85-93])和求解cauchy型积分的快速多极方法(参见[V.]。J.罗克林。理论物理。， 60 (1985)， pp. 187-207]， [L]。格林加德和罗克林，J.康普特。理论物理。[J] .中国农业科学，2003(2)，第3 - 4页。李，L.格林加德，V.罗克林，SIAM J. Sci。中央集权。第一版。， 9(1988)，页669-686]，[L]。F. Greengard，博士论文，耶鲁大学计算机科学系，纽黑文，CT, 1987])。先前发表的通过Kerzman-Stein方程构造保形映射的方法具有O(n^2)$阶的渐近CPU时间估计，其中n是被映射区域边界离散化中的节点数。本文提出的方法具有O(n)阶的估计，使其成为许多情况下的一种选择方法。通过数值算例说明了该算法的性能。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Siam Journal on Scientific and Statistical Computing

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