A Newton method for harmonic mappings in the plane
| Autor: | Sète, Olivier, Zur, Jan |
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| Rok vydání: | 2019 |
| Předmět: | |
| Zdroj: | IMA Journal of Numerical Analysis, Volume 40(4), 2020, pp. 2777-2801 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1093/imanum/drz042 |
| Popis: | 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. Comment: 26 pages, 10 figures. Improved visualization of the dynamics of the harmonic Newton map. Some minor further improvements |
| Databáze: | arXiv |
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