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A charge simulation method for the numerical conformal mapping of interior, exterior and doubly-connected domains. (English) Zbl 0818.30004
It is well known that the conformal mapping of a simply [or doubly] connected domain $D$ onto the unit disk [or a circular ring] can be reduced to the solution of a Dirichlet problem. The author solves the latter by the `charge simulation method’ which has been studied extensively by several Japanese authors: Murashima, Katsurada, Okamoto and the present author. In this method the solution of the Dirichlet problem is approximated by $G(z)= \sum\sp N\sb{i= 1} Q\sb i\log \vert z- \zeta\sb i\vert$ with unknown coefficients $Q\sb i$, where the charge points $\zeta\sb i$ lie outside $\overline D$. To determine the $Q\sb i$ it is required that $G(z\sb j)= b(z\sb j)$ for $N$ collocation points $z\sb j\in \partial D$. The error is defined by $E\sb G= \max\vert G(z\sb{j+{1\over 2}})- b(z\sb{j+{1\over 2}})\vert$, where $z\sb{j+{1\over 2}}\in \partial D$ are intermediate points. The determination of $G$ thus leads to the solution of a $N\times N$ linear system for the $Q\sb i$. -- The paper fully covers earlier work and gives a unified treatment of three types of mapping problems. The choice of the charge points is discussed, and the condition of the $N\times N$ matrix is studied. Results of 11 numerical examples are given and compared with previous experiments. Good results are reported in all cases provided that no concave corners on $\partial D$ occur.

MSC:
30C30Numerical methods in conformal mapping theory
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Full Text: DOI
References:
[1] Amano, K.: Numerical conformal mapping based on the charge simulation method. Trans. inform. Process. soc. Japan 28, 697-704 (1987)
[2] Amano, K.: Numerical conformal mapping of exterior domains based on the charge simulation method. Trans. inform. Process. soc. Japan 29, 62-72 (1988)
[3] Amano, K.: Numerical conformal mapping of doubly-connected domains by the charge simulation method. Trans. inform. Process. soc. Japan 29, 914-924 (1988)
[4] Amano, K.: A bidirectional method for numerical conformal mapping based on the charge simulation method. J. inform. Process. 14, 473-482 (1991) · Zbl 0769.30003
[5] Berrut, J. -P.: A Fredholm integral equation of the second kind for conformal mapping. Numerical conformal mapping, 99-110 (1986) · Zbl 0577.30010
[6] Fairweather, G.; Johnston, R. L.: The method of fundamental solutions for problems in potential theory. Treatment of integral equations by numerical methods, 349-359 (1982) · Zbl 0511.65087
[7] Gaier, D.: Konstruktive methoden der konformen abbildung. (1964) · Zbl 0132.36702
[8] Gaier, D.: Integralgleichungen erster art und konforme abbildung. Math. Z. 147, 113-129 (1976) · Zbl 0304.30006
[9] Gaier, D.: Das logarithmische potential und die konforme abbildung mehrfach zusammenhängender gebiete. E.B. Christoffel, the influence of his work on mathematics and the physical sciences, 290-303 (1981)
[10] Hayes, J. K.; Kahaner, D. K.; Kellner, R. G.: An improved method for numerical conformal mapping. Math. comp. 26, 327-334 (1972) · Zbl 0239.65033
[11] Henrici, P.: Fast Fourier methods in computational complex analysis. SIAM rev. 21, No. 4, 481-527 (1979) · Zbl 0416.65022
[12] Henrici, P.: Applied and computational complex analysis. 3 (1986) · Zbl 0578.30001
[13] Hough, D. M.; Papamichael, N.: The use of splines and singular functions in an integral equation method for c conformal mapping. Numer. math. 37, 133-147 (1981) · Zbl 0441.30016
[14] Hough, D. M.; Papamichael, N.: An integral equation method for the numerical conformal mapping of interior, exterior and doubly-connected domains. Numer. math. 41, 287-307 (1983) · Zbl 0489.30008
[15] Katsurada, M.: A mathematical study of the charge simulation method II. J. fac. Sci. univ. Tokyo sect. IA math. 36, 135-162 (1989) · Zbl 0681.65081
[16] Katsurada, M.: Asymptotic error analysis of the charge simulation method in a Jordan region with an analytic boundary. J. fac. Sci. univ. Tokyo sect. IA math. 37, 635-657 (1990) · Zbl 0723.65093
[17] Katsurada, M.; Okamoto, H.: A mathematical study of the charge simulation method I. J. fac. Sci. univ. Tokyo sect. IA math. 35, 507-518 (1988) · Zbl 0662.65100
[18] Kitagawa, T.: On the numerical stability of the method of fundamental solution applied to the Dirichlet problem. Japan J. Appl. math. 5, 123-133 (1988) · Zbl 0644.65060
[19] Mathon, R.; Johnston, R. L.: The approximate solution of elliptic boundary-value problems by fundamental solutions. SIAM J. Numer. anal. 14, 638-650 (1977) · Zbl 0368.65058
[20] Murashima, S.: Charge simulation method and its application. (1983) · Zbl 0541.73098
[21] Murashima, S.; Kuhara, H.: An approximate method to solve two-dimensional Laplace’s equation by means of superposition of Green’s functions on a Riemann surface. J. inform. Process. 3, 127-139 (1980) · Zbl 0444.65079
[22] Reichel, L.: A fast method for solving certain integral equations of the first kind with application to conformal mapping. Numerical conformal mapping, 125-142 (1986) · Zbl 0587.30007
[23] Singer, H.; Steinbigler, H.; Weiss, P.: A charge simulation method for the calculation of high voltage fields. IEEE trans. Power apparatus systems, 1660-1668 (1974)
[24] Steinbigler, H.: Anfangsfeldstärken und ausnutzungsfaktoren rotationssymmetrischer elektrodenanordnungen in luft. Doctoral thesis (1969)
[25] Symm, G. T.: An integral equation method in conformal mapping. Numer. math. 9, 250-258 (1966) · Zbl 0156.16901
[26] Symm, G. T.: Numerical mapping of exterior domains. Numer. math. 10, 437-445 (1967) · Zbl 0155.21502
[27] Symm, G. T.: Conformal mapping of doubly-connected domains. Numer. math. 13, 448-457 (1969) · Zbl 0174.20602
[28] Trefethen, L. N.: Numerical conformal mapping. (1986) · Zbl 0572.00018