On a~problem of the constructive theory of harmonic mappings

S. I. Bezrodnykh ; V. I. Vlasov

Contemporary Mathematics. Fundamental Directions, Proceedings of the Sixth International Conference on Differential and Functional-Differential Equations (Moscow, August 14–21, 2011). Part 2, Tome 46 (2012), pp. 5-30.

Résumé

The problem of irremovable error appears in finite difference realization of the Winslow approach in the constructive theory of harmonic mappings. As an example, we consider the well-known Roache–Steinberg problem and demonstrate a new approach, which allows us to construct harmonic mappings of complicated domains effectively and with high precision. This possibility is given by the analytic-numerical method of multipoles with exponential convergence rate. It guarantees effective construction of a harmonic mapping with precision controlled by an a posteriori estimate in a uniform norm with respect to the domain.

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