Numerical conformal mapping methods based on Faber series
DeLillo, Thomas K. Elcrat, Alan R. Pfaltzgraff, J. A.
DeLillo, Thomas K.
Elcrat, Alan R.
Pfaltzgraff, J. A.
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1997-10-07
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Keywords
Crowding,Faber series,Fornberg's methods,Numerical conformal mapping,Approximation theory,Boundary conditions,Computational methods,Fast fourier transforms,Iterative methods,Mathematical operators,Discretization,Conformal mapping
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Citation
Thomas K. DeLillo, Alan R. Elcrat, John A. Pfaltzgraff, Numerical conformal mapping methods based on Faber series, Journal of Computational and Applied Mathematics, Volume 83, Issue 2, 1997, Pages 205-236, ISSN 0377-0427, https://doi.org/10.1016/S0377-0427(97)00099-X.
Abstract
Methods are presented for approximating the conformal map from the interior of various regions to the interior of simply-connected target regions with a smooth boundary. The methods for the disk due to Fornberg (1980) and the ellipse due to DeLillo and Elcrat (1993) are reformulated so that they may be extended to other new computational regions. The case of a cross-shaped region is introduced and developed. These methods are used to circumvent the severe ill-conditioning due to the crowding phenomenon suffered by conformal maps from the unit disk to target regions with elongated sections while preserving the fast Fourier methods available on the disk. The methods are based on expanding the mapping function in the Faber series for the regions. All of these methods proceed by approximating the boundary correspondence of the map with a Newton-like iteration. At each Newton step, a system of linear equations is solved using the conjugate gradient method. The matrix-vector multiplication in this inner iteration can be implemented with fast Fourier transforms at a cost of O(N log N). It is shown that the linear systems are discretizations of the identity plus a compact operator and so the conjugate gradient method converges superlinearly. Several computational examples are given along with a discussion of the accuracy of the methods. © 2017 Elsevier B.V., All rights reserved.
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This is an open access article under the CC BY license.
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Elsevier B.V.
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Journal of Computational and Applied Mathematics
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03770427