A Newton method for harmonic mappings in the plane

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 $f = h + \bar{g}$ 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.