Numerical methods for Riemann–Hilbert problems in multiply connected circle domains

Riemann–Hilbert problems in multiply connected domains arise in a number of applications, such as the computation of conformal maps. As an example here, we consider a linear problem for computing the conformal map from the exterior of mm disks to the exterior of mm linear slits with prescribed inclinations. The map can be represented as a sum of Laurent series centered at the disks and satisfying a certain boundary condition. R. Wegmann developed a method of successive conjugation for finding the Laurent coefficients. We compare this method to two methods using least squares solutions to the problem. The resulting linear system has an underlying structure of the form of the identity plus a low rank operator and can be solved efficiently by conjugate gradient-like methods.