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