The treatment of corner and pole-type singularities in numerical conformal mapping techniques

Abstract

This paper is a report of recent developments concerning the nature and the treatment of singularities that affect certain numerical conformal mapping techniques. The paper also includes some new results on the nature of singularities that the mapping function may have in the complement of the closure of the domain under consideration.

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

“…We also consider how the use of "augmented" basis sets, of the type considered in [10] and [13][14][15][16][17][18], affect the stability and convergence properties of the two methods. These augmented sets are formed by introducing into the monomial sets (1.11) and (1.12) "singular" functions that reflect the main singular behaviour of the conformal maps on Of 2 and in compl(f2~Of2).…”

“…That is, we consider the case where the basis set is formed by introducing into one of the monomial sets (3.1) or (3.15) a fixed number m of "singular" functions of the type used in [10,[13][14][15][16][17][18]. As before, we denote the basis set by {t/j} and assume that, corresponding to the ordering t/l, t/2 .... , the m singular functions are t/sx, t/s2, ..-, qsm" (3.25)…”

$L^2$-Approximations of power and logarithmic functions with applications to numerical conformal mapping

“…This was first observed by Levin, Papamichael and Sideridis [14] and subsequently used by Papamichael, Kokkinos, Hough and Warby for improving the convergence rates of certain orthonormalization methods associated with the mapping of interior, exterior and doublyconnected domains; see e.g. [16], [15], [19] and [18].…”

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

“…Further details concerning the construction of appropriate basis sets for dealing with corner and other singularities in expansion methods for numerical conformal mapping can be found in [8,10,12,15]. Here, we only note the following in connection with the use of the ONM.…”

“…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.…”