{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:35:05Z","timestamp":1787337305538,"version":"3.56.0"},"reference-count":27,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>We address the problem of conformally mapping the unit disk to polygons with elongations. The elongations cause the derivative of the conformal map to be exponentially large in some regions. This crowding phenomenon creates difficulties in standard numerical methods for the computation of the conformal map. We make use of the Schwarz\u2013Christoffel representation of the mapping and show that a simple change to the existing algorithms introduced by Trefethen [SIAM J. Sci. Statist. Comput., 1 (1980), pp. 82\u2013102] makes it feasible to accurately compute conformal maps to polygons even in the presence of extreme crowding. For an efficient algorithm it is essential that a good initial guess for the parameters of the Schwarz\u2013Christoffel map be available. A uniformly close initial guess can be obtained from the cross-ratios of certain quadrilaterals, as introduced in the CRDT algorithm of Driscoll and Vavasis [SIAM J. Sci. Comput., 19 (1998), pp. 1783\u20131803]. We present numerical experiments and compare our algorithms with the CRDT which has been particularly designed to combat crowding.<\/jats:p>","DOI":"10.1137\/060677392","type":"journal-article","created":{"date-parts":[[2008,2,14]],"date-time":"2008-02-14T11:55:06Z","timestamp":1202990106000},"page":"618-636","source":"Crossref","is-referenced-by-count":16,"title":["Revisiting the Crowding Phenomenon in Schwarz\u2013Christoffel Mapping"],"prefix":"10.1137","volume":"30","author":[{"given":"L.","family":"Banjai","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,2,14]]},"reference":[{"key":"R1","unstructured":"L. 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Davis,\n                      Numerical methods for coordinate generation based on Schwarz-Christoffel transformations\n                      , in 4th Proceedings of the AIAA Computational Fluid Dynamics Conference, Williamsburg, VA, 1979, AIAA, Rester, VA, pp. 1\u201315.","DOI":"10.2514\/6.1979-1463"},{"key":"R7","doi-asserted-by":"crossref","unstructured":"J. E. Dennis, Jr., and R. B. Schnabel,\n                      Numerical Methods for Unconstrained Optimization and Nonlinear Equations\n                      , Classics in Appl. Math. 16, SIAM, Philadelphia, PA, 1996.","DOI":"10.1137\/1.9781611971200"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1145\/229473.229475"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1145\/1067967.1067971"},{"key":"R10","doi-asserted-by":"crossref","unstructured":"T. A. Driscoll and L. N. 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Nehari,\n                      Conformal Mapping\n                      , McGraw\u2013Hill, New York, Toronto, London, 1952."},{"key":"R22","doi-asserted-by":"crossref","unstructured":"M. L. Overton,\n                      Numerical Computing with IEEE Floating Point Arithmetic\n                      , SIAM, Philadelphia, PA, 2001.","DOI":"10.1137\/1.9780898718072"},{"key":"R23","doi-asserted-by":"crossref","unstructured":"C. Pommerenke,\n                      Boundary Behaviour of Conformal Maps\n                      , Vol. 299, Springer-Verlag, Berlin, 1992.","DOI":"10.1007\/978-3-662-02770-7"},{"key":"R24","unstructured":"R. Schinzinger and P. A. A. Laura,\n                      Conformal Mapping: Methods and Applications\n                      , Elsevier Science, Amsterdam, 1991."},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1137\/0901004"},{"key":"R26","unstructured":"L. N. Trefethen,\n                      SCPACK User's Guide\n                      , MIT Numerical Analysis Report 89-2, MIT Press, Cambridge, MA, 1989."},{"key":"R27","unstructured":"J. H. Wilkinson,\n                      Rounding Errors in Algebraic Processes\n                      , Prentice\u2013Hall, Englewood Cliffs, NJ, 1963."}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/060677392","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:55:55Z","timestamp":1787334955000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/060677392"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,1]]},"references-count":27,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2008,1]]}},"alternative-id":["10.1137\/060677392"],"URL":"https:\/\/doi.org\/10.1137\/060677392","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,1]]}}}