A conformal map formula for difficult cases
The standard problem for a difficult case is the conformal mapping of the set of all points in \({\mathbb{R}}^ 2\) (called z-plane) above a given boundary curve onto the upper half-plane of \({\mathbb{R}}^ 2\) (called w- plane). By applying exp(iz), exp(iw), respectively this problem can be transferred to a mapping between a simply connected region and the unit disk. The difficulties arise either when the given boundary curve varies considerably or when the given simply connected region is ''long''. The author mentions as examples a region bounded below by \(y(x)=-c \cos x\) with c large, say \(c=100\), or ellipses with large ratio c of large and small axis. In all these cases ''crowding'' is an apparent feature. That means that portions of the boundary are enormously compressed by the mapping. This compression factor in the mentioned examples is exponential in c. In an earlier paper [\textit{R. Menikoff} and the author, J. Comput. Phys. 36, 366-410 (1986; Zbl 0434.30007)] the integral equation \[ (*)\quad y(x)=const+\frac{1}{\pi}\int^{\pi}_{-\pi}\log | \sin \frac{1}{2}(u(x)-u(\quad x'))| dx' \] is derived, where y represents the given boundary and u the wanted boundary correspondence. The author calls a boundary point (x,y(x)) crowded, if the (unknown) compression factor \(du/dx<1\), he calls it severely crowded if \(du/dx\ll 1\). Several considerations lead the author to the simplified, approximate compression \[ (**)\quad \frac{\sin \{0.5(u(x)-u_ 0)\}}{\sin \{0,5(u(x_ 1)-u_ 0)\}}=\frac{\sinh Q(x)\quad}{\sinh Q(x_ 1)} \] for the unknown boundary correspondence u, where Q can be expressed in terms of the known y, \(u_ 0=u(x_ 0)\), \(x_ 0\) a minimum point of y (with high compression) and x, \(x_ 1\) severely crowded points near \(x_ 0\). Thus (**) is a local (asymptotic) formula. There are several numerical results including comparisons with the exact mapping which show that calculations of u based on (**) yield results such that the ''crowding'' du/dx has the right order of magnitude. Besides \(y(x)=-c \cos x\) the numerical examples include several periodic y having spikes and singularities. Also the mapping of a (long) ellipse onto the disk is included. Since the method seems to work well, theoretical investigations (convergence, convergence rates, error estimates) are encouraged.
scientific article; zbMATH DE number 3946525