Numerical conformal mapping via the Bergman kernel using the generalized minimum residual method

The Bergman kernel function is known to satisfy a certain boundary integral equation of the second kind. For boundaries that possess symmetrical qualities, the integral equation can be transformed into another integral equation that uses only a small part of the original boundary. This paper applies an iterative procedure known as the generalized minimum residual method for the computation of the Riemann mapping function via the Bergman kernel. The complexity of this procedure is O(n2), where n is the number of collocation points on the boundary of the region. Numerical implementation on some test regions is also presented.