Nasser, Mohamed M. S.; Vuorinen, Matti Conformal invariants in simply connected domains. (English) Zbl 1459.65035 Comput. Methods Funct. Theory 20, No. 3-4, 747-775 (2020). Summary: This paper studies the numerical computation of several conformal invariants of simply connected domains in the complex plane including, the hyperbolic distance, the reduced modulus, the harmonic measure, and the modulus of a quadrilateral. The used method is based on the boundary integral equation with the generalized Neumann kernel. Several numerical examples are presented. The performance and accuracy of the presented method is validated by considering several model problems with known analytic solutions. Cited in 6 Documents MSC: 65E05 General theory of numerical methods in complex analysis (potential theory, etc.) 65E10 Numerical methods in conformal mappings 30C85 Capacity and harmonic measure in the complex plane 31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions 30C30 Schwarz-Christoffel-type mappings Keywords:conformal mappings; hyperbolic distance; reduced modulus; harmonic measure; quadrilateral domains Software:Schwarz-Christoffel; Algorithm 788; SC Toolbox; GitHub; CircularMap; FMMLIB2D PDFBibTeX XMLCite \textit{M. M. S. Nasser} and \textit{M. Vuorinen}, Comput. Methods Funct. Theory 20, No. 3--4, 747--775 (2020; Zbl 1459.65035) Full Text: DOI arXiv References: [1] Ahlfors, LV, Conformal Invariants (1973), New York: McGraw-Hill, New York · Zbl 0272.30012 [2] Anderson, GD; Vamanamurthy, MK; Vuorinen, M., Conformal Invariants. Inequalities and Quasiconformal Maps (1997), New York: Wiley-Interscience, New York · Zbl 0885.30012 [3] Atkinson, K.; Jeon, Y., Algorithm 788: Automatic boundary integral equation program for the planar Laplace equation, Electron. ACM Trans. Math. Softw., 24, 4, 395-417 (1998) · Zbl 0934.65131 · doi:10.1145/293686.293692 [4] Beardon, AF, The Geometry of Discrete Groups (1983), New York: Springer, New York · Zbl 0528.30001 [5] Driscoll, T.A.: Schwarz-Christoffel Toolbox, Version 2.4.1, http://github.com/tobydriscoll/sc-toolbox. Accessed 3 2019 (2019) · Zbl 0884.30005 [6] Driscoll, TA; Trefethen, LN, Schwarz-Christoffel Mapping (2002), Cambridge: Cambridge University Press, Cambridge · Zbl 1003.30005 [7] Dubinin, VN, Condenser Capacities and Symmetrization in Geometric Function Theory (2014), Basel: Springer, Basel · Zbl 1305.30002 [8] Garnett, JB; Marshall, DE, Harmonic Measure (2008), Cambridge: Cambridge University Press, Cambridge [9] Greengard, L., Gimbutas, Z.: FMMLIB2D: a MATLAB toolbox for fast multipole method in two dimensions, Version 1.2, http://www.cims.nyu.edu/cmcl/fmm2dlib/fmm2dlib.html. Accessed 1 2018 (2018) [10] Kanas, S.; Sugawa, T., On conformal representations of the interior of an ellipse, Ann. Acad. Sci. Fenn. Math., 31, 329-348 (2006) · Zbl 1098.30011 [11] Keen, L.; Lakic, N., Hyperbolic Geometry From a Local Viewpoint (2007), Cambridge: Cambridge University Press, Cambridge · Zbl 1190.30001 [12] von Koppenfels, W.; Stallman, F., Praxis der konformen Abbildung (1959), Berlin: Springer, Berlin · Zbl 0086.28003 [13] Lehto, O.; Virtanen, KI, Quasiconformal Mappings in the Plane (1973), Berlin: Springer, Berlin · Zbl 0267.30016 [14] Liesen, J.; Séte, O.; Nasser, MMS, Fast and accurate computation of the logarithmic capacity of compact sets, Comput. Methods Funct. Theory, 17, 689-713 (2017) · Zbl 1381.65026 · doi:10.1007/s40315-017-0207-1 [15] Menikoff, R.; Zemach, C., Methods for numerical conformal mapping, J. Comput. Phys., 36, 366-410 (1980) · Zbl 0434.30007 · doi:10.1016/0021-9991(80)90166-7 [16] Nasser, MMS, Numerical conformal mapping via a boundary integral equation with the generalized Neumann kernel, SIAM J. Sci. Comput., 31, 1695-1715 (2009) · Zbl 1198.30009 · doi:10.1137/070711438 [17] Nasser, MMS, Fast solution of boundary integral equations with the generalized Neumann kernel, Electron. Trans. Numer. Anal., 44, 189-229 (2015) · Zbl 1330.65185 [18] Nasser, MMS, Fast computation of the circular map, Comput. Methods Funct. Theory, 15, 187-223 (2015) · Zbl 1318.30013 · doi:10.1007/s40315-014-0098-3 [19] Nasser, MMS; Murid, AHM; Zamzamir, Z., A boundary integral method for the Riemann-Hilbert problem in domains with corners, Complex Var. Elliptic Equ., 53, 989-1008 (2008) · Zbl 1159.30023 · doi:10.1080/17476930802335080 [20] Nasser, MMS; Vuorinen, M., Computation of conformal invariants, Appl. Math. Comput., 389, 125617 (2021) · Zbl 1474.65060 [21] Papamichael, N.; Stylianopoulos, N., Numerical Conformal Mapping. Domain Decomposition and the Mapping of Quadrilaterals (2010), Hackensack: World Scientific, Hackensack · Zbl 1213.30003 [22] Tsuji, M., Potential Theory in Modern Function Theory (1975), New York: Chelsea Publ. Co., New York · Zbl 0322.30001 [23] Vasil’ev, A., Moduli of Families of Curves for Conformal and Quasiconformal Mappings (2002), Berlin: Springer, Berlin · Zbl 0999.30001 [24] Vuorinen, M., Conformal Geometry and Quasiregular Mappings (1988), Berlin: Springer, Berlin · Zbl 0646.30025 [25] Wegmann, R.; Murid, AHM; Nasser, MMS, The Riemann-Hilbert problem and the generalized Neumann kernel, J. Comput. Appl. Math., 182, 388-415 (2005) · Zbl 1070.30017 · doi:10.1016/j.cam.2004.12.019 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.