N. Papamichael

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

Abstract Let F be the function which maps conformally a simply-connected domain Ω onto a rectangle R, so that four specified points on ∂Ω are mapped 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.

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

SummaryA numerical method, based on the integral equation formulation of Symm, is described for computing approximations to the mapping functions which accomplish the following conformal maps: (a) the mapping of a domain interior to a closed Jordan curve onto the interior of the unit disc, (b) the mapping of a domain exterior to a closed Jordan curve onto the exterior of the unit disc, (c) the mapping of a doubly-connected domain bounded by two closed Jordan curves onto a circular annulus. The numerical method is based on approximating the unknown source density by cubic splines and “singular” functions, and is particularly suited for the mapping of difficult domains having sharp corners.

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

SummaryIn 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.

The use of splines and singular functions in an integral equation method for conformal mapping

SummaryWe consider the integral equation method of Symm for the conformal mapping of simply-connected domains. For the numerical solution, we examine the use of spline functions of various degrees for the approximation of the source density σ. In particular, we consider ways for overcoming the difficulties associated with corner singularities. For this we modify the spline approximation and in the neighborhood of each corner, where a boundary singularity occurs, we approximate σ by a function which reflects the main singular behaviour of the source density. The singular functions are then blended with the splines, which approximate σ on the remainder of the boundary, so that the global approximating function has continuity of appropriate order at the transition points between the two types of approximation. We show, by means of numerical examples, that such approximations overcome the difficulties associated with corner singularities and lead to numerical results of high accuracy.

A cubic spline method for the solution of a linear fourth-order two-point boundary value problem

Abstract A cubic spline method is described for the numerical solution of a two-point boundary value problem, involving a fourth order linear differential equation. This spline method is shown to be closely related to a known fourth order finite difference scheme.

Numerical techniques for conformal mapping onto a rectangle

Abstract This paper is concerned with the problem of determining approximations to the function F which maps conformally a simply-connected domain Ω onto a rectangle R, so that four specified points on ∂Ω are mapped respectively onto the four vertices of R. In particular, we study the following two classes of methods for the mapping of domains of the form Ω≔ {z = x + iy:00 1 (x) 2 (x)} . (i) Methods which approximate F: Ω → R by F = S ∘ F , where F is an approximation to the conformal map of Ω onto the unit disc, and S is a simple Schwarz-Christoffel transformation. (ii) Methods based on approximating the conformal map of a certain symmetric doubly-connected domain onto a circular annulus.

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.

A Domain Decomposition Method for Conformal Mapping onto a Rectangle

Letg be the function which maps conformally a rectangleR onto a simply connected domainG so that the four vertices ofR are mapped respectively onto four specified pointsz1,z2,z3,z4 on∂G. This paper is concerned with the study of a domain decomposition method for computing approximations tog 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 pointsz1,z2,z3,Z4, are the corners where the two straight lines meet the two arcs.

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

SummaryThis 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 quadrilaterals than that indicated above.

On the theory and application of a domain decomposition method for computing conformal modules

We consider the theory and application of a domain decomposition method for computing the conformal modules of long quadrilaterals. The method has been studied already by us and also by Gaier and Hayman. Our main purpose here is to extend its area of application and, in the same time, improve some of our earlier error estimates.