Numerical computation of the Ahlfors map of a multiply connected planar domain

Abstract

N. Kerzman and E. M. Stein discovered in [6] a method for computing the SzegG Kernel of a bounded domain D in the complex plane with C” smooth boundary. In case D is also simply connected, the Kerzman-Stein method yields a powerful technique for computing the Riemann Mapping Function associated to a point a in D (see [6, 71). In this note, we show how the Kerzman-Stein method can be generalized to yield a method for computing the Ahlfors map associated to a point in a finitely connected, bounded domain in the plane with C* smooth boundary. The Ahlfors map is a proper holomorphic mapping of D onto the unit disc which maps each boundary component of D one-to-one onto the boundary of the unit disc. The Ahlfors map might prove to be useful in certain problems arising in fluid mechanics. For example, in the problem of computing the transonic flow past an obstacle in the plane, a conformal map of the outside of the obstacle onto the unit disc is used to set up a grid which is most convenient for making numerical computations (see [S] ). The Ahlfors map could be used in a similar way in problems of this sort in which more than one obstacle is involved. These and other applications will be explored in subsequent papers. For the present, we content ourselves with describing our numerical method and proving its validity. It should be mentioned that in case D is simply connected, the Ahlfors map is equal to the Riemann Mapping Function. Thus, our numerical method yields a new technique for computing the Riemann map. In this setting, however, the method of [7] is to be preferred.