Conformal mapping by the method of alternating projections
The conformal mapping function \(\Phi\) of the unit disc onto a region G with \(0\in G\) normalized by \(\Phi (0)=0\), \(\Phi '(0)>0\), is characterized by two properties: (1) \(\Phi\) is analytic and satisfies \(\Phi (0)=0\), Im \(\Phi\) '(0)\(=0\), (2) \(\Phi (e^{it})\in \partial G\). For regions with smooth boundaries the functional analytic principle of alternating projections applied to certain manifolds in \(L^ 2\) can be used to construct a sequence of functions which alternatingly satisfy one of these properties. A theorem of Ostrowski can be used to prove local linear convergence in a Sobolev space. The limit function necessarily has both properties (1) and (2) and therefore is equal to \(\Phi\). A version with overrelaxation as well as a discretized version are discussed along the same lines. The resulting iterative method is very simple. It requires in each step only two complex Fourier transforms. It can be implemented efficiently using FFT. The convergence factor is equal to the spectral radius of a certain operator, which can be calculated approximately for several of the standard regions. For regions close to the unit disc convergence of the overrelaxed method is very fast.
scientific article; zbMATH DE number 4087964