Numerical conformal mapping of exterior domains

Abstract

w926004× SUMMARYThe work of the present paper is closely related to the two numerical procedures described in [11], for determining approximations to the function which maps conformally a bounded simply-connected domain Ω 1 , with boundary ∂Ω, onto the unit disc. Here, we consider the use of these procedures for the solution of the corresponding exterior problem, i.e. the problem of determining approximations to the mapping function which maps conformally the exterior domain Ω = compl(Ω I ⋃∂Ω) onto the unit disc.

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

“…In general however the situation regarding the convergence of approximations obtained by using an augmented basis is not clear. For example, the numerical experiments of [10] and [13,14] indicate clearly that substantial improvement in accuracy is achieved when, in the presence of a "singular" corner at ZoeOf2, the basis set contains functions of the form (z -Zo) ~ {Log (z -z0)} m,…”

“…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).…”

“…5.2, the mapping function fez is not known exactly. For this reason, the values of _R2 a/2 listed in the table are estimates computed by using BKM/AB approximations to fez; see [14,Ex. 3.3].…”

“…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)…”

An integral equation method for the numerical conformal mapping of interior, exterior and doubly-connected domains

“…These values are accurate approximations to c and M, computed in [10] and [11] respectively by using certain variational and orthonormalization methods, based on the theory of Bergman kernel functions.…”