A numerical method for the conformal mapping of simply-connected d o ma i n s o n t o t h e u n i t d i s c i s c o n s i d e r e d . T h e me t h o d i s b a s e d on the use of the Bergman kernel function of the domain. It is s h o w n t h a t , f o r a s u c c e s s f u l a p p l i c a t i o n , t h e b a s i s o f t h e s e r i e s representation of the kernel must include terms that reflect the main singular behaviour of the kernel in the complement of the domain.2

w9260228 SUMMARYTwo numerical methods are considered for the conformal mapping of a bounded simply-connected domain onto the unit disc. The two methods are respectively the Bergman kernel method, which has been described in [17], and the so-called Ritz method.In this paper we indicate the close theoretical relationship of the two methods, compare their computational efficiencies and present a number of practical applications of the approximate conformal maps.

Summary. In this paper we study the stability and convergence properties of Bergman kernel methods, for the numerical conformal mapping of simply and doubly-connected domains. In particular, by using certain wellknown results of Carleman, we establish a characterization of the level of instability in the methods, in terms of the geometry of the domain under consideration. We also explain how certain known convergence results can provide some theoretical justification of the observed improvement in accuracy which is achieved by the methods, when the basis set used contains functions that reflect the main singular behaviour of the conformal map.

Let g be the function which maps conformally a rectangle R onto a simply connected domain G so that the four vertices of R are mapped respectively onto four specified points z I, z 2, za, z4 on tgG. This paper is concerned with the study of a domain decomposition method for computing approximations to g and to an associated domain functional in cases where: (i) G is bounded by two parallel straight lines and two Jordan arcs. (ii) The four points zl, z2, z3, z4, are the corners where the two straight lines meet the two arcs.

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