Numerical conformal mapping based on the generalised conjugation operator

An iterative procedure for numerical conformal mapping is presented which imposes no restriction on the boundary complexity. The formulation involves two analytically equivalent boundary integral equations established by applying the conjugation operator to the real and the imaginary parts of an analytical function. The conventional approach is to use only one and ignore the other equation. However, the discrete version of the operator using the boundary element method (BEM) leads to two non-equivalent sets of linear equations forming an over-determined system. The generalised conjugation operator is introduced so that both sets of equations can be utilised and their least-square solution determined without any additional computational cost, a strategy largely responsible for the stability and efficiency of the proposed method. Numerical tests on various samples including problems with cracked domains suggest global convergence, although this cannot be proved theoretically. The computational efficiency appears significantly higher than that reported earlier by other investigators.

     

Numerical conformal mapping of multiply connected regions by Fornberg-like methods

Summary. We develop a new algorithm for computing conformal maps from regions exterior to non-overlapping disks to unbounded multiply connected regions exterior to non-overlapping, smoothly bounded Jordan regions. The method is an extension of Fornberg's original Newton-like method for mapping of the disk to simply connected regions. A Fortran program based on the algorithm has been developed and tested for the 2 and 3 disk case. Numerical examples are reported. Received March 12, 1998 / Revised version received December 16, 1998     

Branch point area methods in conformal mapping

The classical estimate of Bieberbach that ?a ₂?≤2 for a given univalent function ?(z)=z+a ₂ z ²+… in the classS leads to the best possible pointwise estimates of the ratio ?"(z)/?'(z) for ?∈S, first obtained by K?be and Bieberbach. For the corresponding class Σ of univalent functions in the exterior disk, Goluzin found in 1943 by variational methods the corresponding best possible pointwise estimates of ?"(z)/?'(z) for ψ∈Σ. It was perhaps surprising that this time, the expressions involve elliptic integrals. Here, we obtain an area-type theorem which has Goluzin's pointwise estimate as a corollary. This shows that Goluzin's estimate, like the K?be-Bieberbach estimate, is firmly rooted in areabased methods. The appearance of elliptic integrals finds a natural explanation: they arise because a certain associated covering surface of the Riemann sphere is a torus.     

Osculation methods for the conformal mapping of doubly connected regions

An osculation method for the conformal mapping of a doubly connected region onto an annulus and corresponding numerical experiments are described. The experiments indicate that even difficult mapping problems are solved very efficiently and at low cost.

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Zusammenfassung Für die konforme Abbildung eines zweifach zusammenhängenden Gebiets auf einen Kreisring wird ein Schmiegungsverfahren beschrieben. Die dazu durchgeführten numerischen Experimente ergeben gute Resultate mit geringen Kosten selbst bei komplizierten Abbildungsproblemen.

     

Numerical experiments with a method of successive approximation for conformal mapping

A method of successive approximation for the conformal mapping of simply connected regions onto the unit disc and numerical experiments with it are described. The experiments gave unexpectedly good results.

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Zusammenfassung Es werden ein Schmiegungsverfahren für die konforme Abbildung einfach zusammenhängender Gebiete auf den Einheitskreis und numerische Experimente damit beschrieben. Letztere ergaben unerwartet gute Resultate.

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Born December 6, 1942; killed August 31, 1979, while climbing in the mountains of Canada.     

Faber polynomials corresponding to rational exterior mapping functions

Faber polynomials corresponding to rational exterior mapping functions of degree (m, m − 1) are studied. It is shown that these polynomials always satisfy an (m + 1)-term recurrence. For the special case m = 2, it is shown that the Faber polynomials can be expressed in terms of the classical Chebyshev polynomials of the first kind. In this case, explicit formulas for the Faber polynomials are derived.     

Dzyadyk formula for summation of double Faber series

An analog of the Dzyadyk formula is constructed for double series in Faber polynomials of two variables. By using this formula, we obtain the estimates of the convergence rate in a bicylindrical domain for double Faber series summed over rectangles and circles.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp. 1283–1287, September, 1994.     

Numerical conformal mapping of multiply connected domains to regions with circular boundaries

We propose a method to map a multiply connected bounded planar region conformally to a bounded region with circular boundaries. The norm of the derivative of such a conformal map satisfies the Laplace equation with a nonlinear Neumann type boundary condition. We analyze the singular behavior at corners of the boundary and separate the major singular part. The remaining smooth part solves a variational problem which is easy to discretize. We use a finite element method and a gradient descent method to find an approximate solution. The conformal map is then constructed from this norm function. We tested our algorithm on a polygonal region and a curvilinear smooth region.     

Numerical solution of a problem about the parameters of a conformal mapping

A converging cyclic iteration method is applied for finding the parameters of a conformal mapping of the upper half-plane onto the domain bounded by a given polygon. The numerical realization of the method is presented with a proof of convergence and an estimate of the convergence speed. A few examples of computer calculations is given.     

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

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.     

On semiiterative methods generated by Faber polynomials

Numerische Mathematik - Consider a nonsingular system of linear algebraic equationsA x=b, or in fixed point formx=Tx+c, where the eigenvalues ofT are contained in some compact subset Ω of the...     

Time-dependent conformal mapping of doubly-connected regions

This paper examines two key features of time-dependent conformal mappings in doubly-connected regions, the evolution of the conformal modulus Q(t) and the boundary transformation generalizing the Hilbert transform. It also applies the theory to an unsteady free surface flow. Focusing on inviscid, incompressible, irrotational fluid sloshing in a rectangular vessel, it is shown that the explicit calculation of the conformal modulus is essential to correctly predict features of the flow. Results are also presented for fully dynamic simulations which use a time-dependent conformal mapping and the Garrick generalization of the Hilbert transform to map the physical domain to a time-dependent rectangle in the computational domain. The results of this new approach are compared to the complementary numerical scheme of Frandsen (J. Comput. Phys. 196:53–87, 14) and it is shown that correct calculation of the conformal modulus is essential in order to obtain agreement between the two methods.     

On conformal mapping of polygonal regions

We develop Kufarev's method for determining unknown parameters in the Schwarz-Christoffel integral in the case of conformal mapping of polygonal regions with boundary normalization.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 11, pp. 1484–1494, November, 1993.     

Zolotarev's conformal mapping and Chebotarev's problem

Consider the Chebotarev problem of finding a continuum S in the complex plane including some given points such that the logarithmic capacity of S is minimal. In this paper, we give a complete solution of this problem for the case of three given points with the help of Zolotarev's conformal mapping using Jacobian elliptic and theta functions. Moreover, for four given points, some special cases can be treated.