Rapid methods for the conformal mapping of multiply connected regions

Abstract

We present fast methods for the conformal mapping of simply, doubly and multiply connected regions onto certain canonical regions in the plane. Our mapping procedure consists of two parts. First we solve an integral equation on the boundary of the region we wish to map. The solution of this integral equation is needed to determine the boundary correspondence. We have chosen to use the integral equation formulation of Mikhlin. Although it is not widely used, this formulation has the advantage that it leads to integral equations of the second kind with unique solutions and bounded kernels. The solutions are also periodic, allowing for effective use of the trapezoid rule. Once we have solved the integral equation we use a rapid method we have previously developed to determine the mapping function in the interior of the region. This method makes use of fast Poisson solvers, and thereby circumvents the difficulties associated with computing integrals at points near the boundary of the region, and avoids the expense of computing many integrals. We also provide results of numerical experiments. © 1986.