A Newton method for harmonic mappings in the plane

We present an iterative root finding method for harmonic mappings in the complex plane, which is a generalization of Newton’s method for analytic functions. The complex formulation of the method allows an analysis in a complex variables spirit. For zeros close to poles of $f = h + \overline{g}$ we c...

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Bibliographic Details
Published in:IMA journal of numerical analysis 2020-10, Vol.40 (4), p.2777-2801
Main Authors: Sète, Olivier, Zur, Jan
Format: Article
Language:English
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Summary:We present an iterative root finding method for harmonic mappings in the complex plane, which is a generalization of Newton’s method for analytic functions. The complex formulation of the method allows an analysis in a complex variables spirit. For zeros close to poles of $f = h + \overline{g}$ we construct initial points for which the harmonic Newton iteration is guaranteed to converge. Moreover, we study the number of solutions of $f(z) = \eta $ close to the critical set of $f$ for certain $\eta \in \mathbb{C}$. We provide a MATLAB implementation of the method, and illustrate our results with several examples and numerical experiments, including phase plots and plots of the basins of attraction.
ISSN:0272-4979
1464-3642
DOI:10.1093/imanum/drz042