Discretized versions of Newton type iterative methods for conformal mapping

Abstract

It is shown that for nearly circular regions discretized versions of the Newton type methods of Wegmann (1978) and Hübner (1986) converge locally to fixed points. Convergence is linear. The rates can be determined approximately for several standard regions. The dominant operator acts only on a subspace of high-order harmonics. Therefore under conditions of smoothness and/or symmetry convergence can be much faster. The fixed points satisfy a discrete version of Lavrentev's variational principle. Therefore the resulting approximations for conformal mapping are essentially as accurate as Wittich's approximation for the conjugation operation.