Numerical conformal mapping onto a rectangle with applications to the solution of Laplacian problems

Let F be the function which maps conformally a simple-connected domain onto a rectangle R, so that four specified points on are mapped Ω∂respectively onto the four vertices of R. In this paper we consider the problem of approximating the conformal map F, and present a survey of the available numerical methods. We also illustrate the practical significance of the conformal map, by presenting a number of applications involving the solution of Laplacian boundary value problems

A domain decomposition method for conformal mapping onto a rectangle

Let g be the function which maps conformally a simply-connected domain G onto a rectangle R so that four specified points z1, z2, z3, z4,o n ∂G are mapped respectively onto the four vertices of R. 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 z1, z2, z3, z4, are the corners where the two straight lines meet the two arcs

On the numerical performance of a domain decomposition method for conformal mapping

This paper is a sequel to a recent paper [14], concerning a domain decomposition method (hereafter referred to as DDM ) for the conformal mapping of a certain class of quadrilaterals. For the description of the DDM we proceed exactly as in [14:§1], by introducing the following terminology and notations

End conditions for interpolatory quintic splines

Accurate end conditions are derived for quintic spline interpolation at equally spaced knots. These conditions are in terms of available function values at the knots and lead to 0(h6) covergence uniformly on the interval of interpolation

On the comparison of two numerical methods for conformal mapping

Let G be a simply-connected domain in the t—plane (t = x + iy), bounded by the three straight lines x = 0, y = 0, x =1 and a Jordan arc with cartesian equation y = τ (X). Also, let g be the function which maps conformally a rectangle R onto G, so that the four corners of R are mapped onto those of G. In this paper we show that the method con-sidered recently by Challis and Burley [2], for determining approx- imations to g, is equivalent to a special case of the well-known method of Garrick [8] for the mapping of doubly-connected domains, Hence, by using results already available in the literature, we provide some theoretical justification for the method of [2]

A domain decomposition method for approximating the conformal modules of long quadrilaterals

This paper is concerned with the study of a domain decomposition method for approximating the conformal modules of long quadrilaterals. The method has been studied already by us and also by D Gaier and W K Hayman, but only in connection with a special class of quadrilaterals, viz. quadrilaterals where: (a) the defining domain is bounded by two parallel straight lines and two Jordan arcs, and (b) the four specified boundary points are the four corners where the arcs meet the straight lines. Our main purpose here is to explain how the method may be extended to a wider class of qua-drilaterals than that indicated above

Improved orders of approximation derived from interpolatory cubic splines

Let s be a cubic spline, with equally spaced knots on [a,b], interpolating a given function y at the knots. The parameters which determine s are used to construct a piecewise defined polynomial P of degree four. It is shown that P can be used to give better orders of approximation to y and its derivatives than those obtained from s. It is also shown that the known superconvergence properties of the derivatives of s, at specific points [a,b], are all special cases of the main result contained in the present paper

The use of singular functions for the approximate conformal mapping of doubly-connected domains

Let f be the function which maps conformally a given doubly- connected domain onto a circular annulus. We consider the use of two closely related methods for determining approximations to f of the form fn (z) = z exp, ⎪⎩⎪⎨⎧⎭⎬⎫Σ−(z)uan1jjj where {uj} is a set of basis functions. The two methods are respectively a variational method, based on an extremum property of the function H(z) = f′(z)/f(z) - 1/z, and an orthononnalization method, based on approximating the function H by a finite Fourier series sum. The main purpose of the paper is to consider the use of the two methods for the mapping of domains having sharp corners, where corner singularities occur. We show, by means of numerical examples, that both methods are capable of producing approximations of high accuracy for the mapping of such "difficult" doubly-connected domains. The essential requirement for this is that the basis set {uj} contains singular functions that reflect the asymptotic behaviour of the function H in the neighbourhood of each "singular" corner