The use of splines and singular functions in an integral equation method for conformal mapping

Summary

We consider the integral equation method of Symm for the conformal mapping of simply-connected domains. For the numerical solution, we examine the use of spline functions of various degrees for the approximation of the source density σ. In particular, we consider ways for overcoming the difficulties associated with corner singularities. For this we modify the spline approximation and in the neighborhood of each corner, where a boundary singularity occurs, we approximate σ by a function which reflects the main singular behaviour of the source density. The singular functions are then blended with the splines, which approximate σ on the remainder of the boundary, so that the global approximating function has continuity of appropriate order at the transition points between the two types of approximation. We show, by means of numerical examples, that such approximations overcome the difficulties associated with corner singularities and lead to numerical results of high accuracy.