A Newton method for harmonic mappings in the plane. (18th November 2019)

• Record Type: Journal Article
• Authors: Sète, Olivier; Zur, Jan
• Is Part Of: IMA journal of numerical analysis. Volume 40:Number 4(2020)
• Page Start: 2777
• Page End: 2801
• Publication Date: 2019-11-18
• Subjects: zeros of harmonic mappings -- Newton's method -- basins of attraction -- domain coloring -- Wilmshurst's conjecture -- gravitational lensing; Numerical analysis -- Periodicals; 519.405
• DOI: 10.1093/imanum/drz042
• Languages: English
• ISSNs: 0272-4979

Abstract

Abstract: 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.