Numerical conformal mapping methods for simply and doubly connected regions

Methods are presented and analyzed for approximating the conformal map from the interior (exterior) of the disk to the interior (exterior) of a smooth, simple closed curve and from an annulus to a bounded, doubly connected region with smooth boundaries. The methods are Newton-like methods for computing the boundary correspondences and conformal moduli similar to Fornberg's method for the interior of the disk. We show that the linear systems are discretizations of the identity plus a compact operator and, hence, that the conjugate gradient method converges superlinearly.