Pole-type singularities and the numerical conformal mapping of doubly-connected domains

Abstract

L e t f b e t h e f u n c t i o n w h i c h m a p s c o n f o r m a l l y a g i v e n d o u b l yconnected domain onto a circular annulus, and let Ω H(z) = f '(z) / f(z)-1/z .

An Orthonormalization Method for the Approximate Conformal Mapping of Multiply-Connected Domains

“…Many numerical examples illustrating the application of the ONM to the mapping of doubly-connected domains, including domains with sharp corners, can be found in references [12,13]. Here, we present six examples whose purpose is to confirm our claim that the ONM can also be used for the mapping of N-connected domains with N>2.…”

“…singularities that occur off the boundary curves in compl(ΩU∂Ω).The problem of dealing with such singularities is studied in [13,15], but only in connection with the doubly-connected case N=2.…”

Stability and convergence properties of Bergman kernel methods for numerical conformal mapping

“…Regarding the use of augmented basis sets, the remarks we made in connection with the BKM also apply to the ONM. In the case of the ONM however, it has been shown in [17] that it is not possible to reflect exactly the singularities of H, even when the singular points are in compl(C2u 00).…”

“…and w=exp(2rci/3). It is also shown in [17] that these singularities can be reflected, but only approximately, by introducing into the monomial set (5.46) functions corresponding to simple poles at the points (5.47). The use of augmented basis sets constructed in this manner leads to improved ONM approximations, especially when the radius a of 0f21 is close to unity.…”

“…However, since the singularities of H at the points (5.47) can only be reflected approximately, we do not have any theoretical results concerning the rate of convergence of the ONM/AB. For this reason, in this example we consider only the use of the ONM/MB and refer the reader to [15] and [17], where several examples involving the use of the ONM with augmented basis sets are considered.…”