Abstract
We present a numerical method for the computation of the conformal map from unbounded multiply-connected domains onto lemniscatic domains. For \(\ell \)-times connected domains, the method requires solving \(\ell \) boundary integral equations with the Neumann kernel. This can be done in \(O(\ell ^2 n \log n)\) operations, where n is the number of nodes in the discretization of each boundary component of the multiply-connected domain. As demonstrated by numerical examples, the method works for domains with close-to-touching boundaries, non-convex boundaries, piecewise smooth boundaries, and for domains of high connectivity.
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Andreev, V.V., McNicholl, T.H.: Computing conformal maps of finitely connected domains onto canonical slit domains. Theory Comput. Syst. 50(2), 354–369 (2012). doi:10.1007/s00224-010-9305-4
Atkinson, K.E.: The numerical solution of integral equations of the second kind. Cambridge Monographs on Applied and Computational Mathematics, vol. 4. Cambridge University Press, Cambridge (1997). doi:10.1017/CBO9780511626340
Austin, A.P., Kravanja, P., Trefethen, L.N.: Numerical algorithms based on analytic function values at roots of unity. SIAM J. Numer. Anal. 52(4), 1795–1821 (2014). doi:10.1137/130931035
Crowdy, D.: The Schwarz–Christoffel mapping to bounded multiply connected polygonal domains. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461(2061), 2653–2678 (2005). doi:10.1098/rspa.2005.1480
Crowdy, D.: Schwarz–Christoffel mappings to unbounded multiply connected polygonal regions. Math. Proc. Camb. Philos. Soc. 142(2), 319–339 (2007). doi:10.1017/S0305004106009832
Crowdy, D., Marshall, J.: Conformal mappings between canonical multiply connected domains. Comput. Methods Funct. Theory 6(1), 59–76 (2006). doi:10.1007/BF03321118
DeLillo, T.K.: Schwarz–Christoffel mapping of bounded, multiply connected domains. Comput. Methods Funct. Theory 6(2), 275–300 (2006). doi:10.1007/BF03321615
DeLillo, T.K., Driscoll, T.A., Elcrat, A.R., Pfaltzgraff, J.A.: Computation of multiply connected Schwarz–Christoffel maps for exterior domains. Comput. Methods Funct. Theory 6(2), 301–315 (2006). doi:10.1007/BF03321616
DeLillo, T.K., Driscoll, T.A., Elcrat, A.R., Pfaltzgraff, J.A.: Radial and circular slit maps of unbounded multiply connected circle domains. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 464(2095), 1719–1737 (2008). doi:10.1098/rspa.2008.0006
DeLillo, T.K., Elcrat, A.R., Kropf, E.H., Pfaltzgraff, J.A.: Efficient calculation of Schwarz–Christoffel transformations for multiply connected domains using Laurent series. Comput. Methods Funct. Theory 13(2), 307–336 (2013). doi:10.1007/s40315-013-0023-1
DeLillo, T.K., Elcrat, A.R., Pfaltzgraff, J.A.: Schwarz–Christoffel mapping of multiply connected domains. J. Anal. Math. 94, 17–47 (2004). doi:10.1007/BF02789040
Delillo, T.K., Kropf, E.H.: Numerical computation of the Schwarz–Christoffel transformation for multiply connected domains. SIAM J. Sci. Comput. 33(3), 1369–1394 (2011). doi:10.1137/100816912
Greenbaum, A., Greengard, L., McFadden, G.B.: Laplace’s equation and the Dirichlet–Neumann map in multiply connected domains. J. Comput. Phys. 105(2), 267–278 (1993). doi:10.1006/jcph.1993.1073
Greengard, L., Gimbutas, Z.: FMMLIB2D: A MATLAB Toolbox for Fast Multipole Method in Two Dimensions, Version 1.2 (2012). http://www.cims.nyu.edu/cmcl/fmm2dlib/fmm2dlib.html
Greengard, L., Rokhlin, V.: A fast algorithm for particle simulations. J. Comput. Phys. 73(2), 325–348 (1987). doi:10.1016/0021-9991(87)90140-9
Grunsky, H.: Über konforme Abbildungen, die gewisse Gebietsfunktionen in elementare Funktionen transformieren. I. Math. Z. 67, 129–132 (1957)
Grunsky, H.: Über konforme Abbildungen, die gewisse Gebietsfunktionen in elementare Funktionen transformieren. II. Math. Z. 67, 223–228 (1957)
Helsing, J., Ojala, R.: On the evaluation of layer potentials close to their sources. J. Comput. Phys. 227(5), 2899–2921 (2008). doi:10.1016/j.jcp.2007.11.024
Henrici, P.: Applied and Computational Complex Analysis, vol. 3. Pure and Applied Mathematics. Wiley, New York (1986)
Jenkins, J.A.: On a canonical conformal mapping of J. L. Walsh. Trans. Am. Math. Soc. 88, 207–213 (1958)
Kaplan, W.: Introduction to Analytic Functions. Addison-Wesley Publishing Co., Reading (1966)
Koch, T., Liesen, J.: The conformal “bratwurst” maps and associated Faber polynomials. Numer. Math. 86(1), 173–191 (2000). doi:10.1007/PL00005401
Koebe, P.: Abhandlungen zur Theorie der konformen Abbildung, IV. Abbildung mehrfach zusammenhängender schlichter Bereiche auf Schlitzbereiche. Acta Math. 41(1), 305–344 (1916). doi:10.1007/BF02422949
Kress, R.: A Nyström method for boundary integral equations in domains with corners. Numer. Math. 58(2), 145–161 (1990). doi:10.1007/BF01385616
Kress, R.: Linear Integral Equations, Applied Mathematical Sciences, vol. 82, 3rd edn. Springer, New York (2014). doi:10.1007/978-1-4614-9593-2
Landau, H.J.: On canonical conformal maps of multiply connected domains. Trans. Am. Math. Soc. 99, 1–20 (1961)
Liesen, J., Sète, O., Nasser, M.M.S.: Computing the Logarithmic Capacity of Compact Sets via Conformal Mapping. arXiv:1507.05793 (2015)
Luo, W., Dai, J., Gu, X., Yau, S.T.: Numerical conformal mapping of multiply connected domains to regions with circular boundaries. J. Comput. Appl. Math. 233(11), 2940–2947 (2010). doi:10.1016/j.cam.2009.11.038
Muskhelishvili, N.I.: Singular Integral Equations. Noordhoff International Publishing, Leyden (1977)
Nasser, M.M.S.: A boundary integral equation for conformal mapping of bounded multiply connected regions. Comput. Methods Funct. Theory 9(1), 127–143 (2009). doi:10.1007/BF03321718
Nasser, M.M.S.: Numerical conformal mapping via a boundary integral equation with the generalized Neumann kernel. SIAM J. Sci. Comput. 31(3), 1695–1715 (2009). doi:10.1137/070711438
Nasser, M.M.S.: Numerical conformal mapping of multiply connected regions onto the second, third and fourth categories of Koebe’s canonical slit domains. J. Math. Anal. Appl. 382(1), 47–56 (2011). doi:10.1016/j.jmaa.2011.04.030
Nasser, M.M.S.: Numerical conformal mapping of multiply connected regions onto the fifth category of Koebe’s canonical slit regions. J. Math. Anal. Appl. 398(2), 729–743 (2013). doi:10.1016/j.jmaa.2012.09.020
Nasser, M.M.S.: Fast computation of the circular map. Comput. Methods Funct. Theory 15(2), 187–223 (2015). doi:10.1007/s40315-014-0098-3
Nasser, M.M.S.: Fast solution of boundary integral equations with the generalized Neumann kernel. Electron. Trans. Numer. Anal. 44, 189–229 (2015)
Nasser, M.M.S., Al-Shihri, F.A.A.: A fast boundary integral equation method for conformal mapping of multiply connected regions. SIAM J. Sci. Comput. 35(3), A1736–A1760 (2013). doi:10.1137/120901933
Nasser, M.M.S., Murid, A.H.M., Ismail, M., Alejaily, E.M.A.: Boundary integral equations with the generalized Neumann kernel for Laplace’s equation in multiply connected regions. Appl. Math. Comput. 217(9), 4710–4727 (2011). doi:10.1016/j.amc.2010.11.027
Nasser, M.M.S., Sakajo, T., Murid, A.H.M., Wei, L.: A fast computational method for potential flows in multiply connected coastal domains. Jpn. J. Ind. Appl. Math. 32(1), 205–236 (2015). doi:10.1007/s13160-015-0168-6
Nehari, Z.: Conformal Mapping. McGraw-Hill Book Co. Inc, New York (1952)
Rathsfeld, A.: Iterative solution of linear systems arising from the Nyström method for the double-layer potential equation over curves with corners. Math. Methods Appl. Sci. 16(6), 443–455 (1993). doi:10.1002/mma.1670160604
Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856–869 (1986). doi:10.1137/0907058
Sète, O., Liesen, J.: On Conformal Maps from Multiply Connected Domains onto Lemniscatic Domains. Electron. Trans. Numer. Anal. 45, 1–15 (2016)
Sète, O., Liesen, J.: Properties and Examples of Faber–Walsh Polynomials. arXiv:1502.07633 (2015)
Walsh, J.L.: On the conformal mapping of multiply connected regions. Trans. Am. Math. Soc. 82, 128–146 (1956)
Walsh, J.L.: A generalization of Faber’s polynomials. Math. Ann. 136, 23–33 (1958)
Walsh, J.L.: Interpolation and approximation by rational functions in the complex domain. American Mathematical Society Colloquium Publications, vol. XX, 5th edn. American Mathematical Society, Providence, RI (1969)
Wegmann, R.: Methods for numerical conformal mapping. In: Handbook of Complex Analysis: Geometric Function Theory, vol. 2, pp. 351–477. Elsevier, Amsterdam (2005). doi:10.1016/S1874-5709(05)80013-7
Wegmann, R., Nasser, M.M.S.: The Riemann–Hilbert problem and the generalized Neumann kernel on multiply connected regions. J. Comput. Appl. Math. 214(1), 36–57 (2008). doi:10.1016/j.cam.2007.01.021
Yunus, A., Murid, A., Nasser, M.: Numerical conformal mapping and its inverse of unbounded multiply connected regions onto logarithmic spiral slit regions and rectilinear slit regions. Proc. R. Soc. A. 470(2162), 514 (2014)
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Communicated by Darren Crowdy.
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Nasser, M.M.S., Liesen, J. & Sète, O. Numerical Computation of the Conformal Map onto Lemniscatic Domains. Comput. Methods Funct. Theory 16, 609–635 (2016). https://doi.org/10.1007/s40315-016-0159-x
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DOI: https://doi.org/10.1007/s40315-016-0159-x
Keywords
- Numerical conformal mapping
- Multiply-connected domains
- Lemniscatic domains
- Boundary integral equations
- Neumann kernel