1,085 research outputs found
Asymptotic behaviour of zeros of bieberbach polynomials
AbstractLet Ω be a simply-connected domain in the complex plane and let πn denote the nth-degree Bieberbach polynomial approximation to the conformal map f of Ω onto a disc. In this paper we investigate the asymptotic behaviour (as n→σ) of the zeros of πn, πn′ and also of the zeroes of certain closely related rational approximants to f. Our result show that, in each case, the distribution of the zeros is governed by the location of the singularities of the mapping function f in C⧹ω, and we present numerical examples illustrating this
A spectral method for elliptic equations: the Dirichlet problem
An elliptic partial differential equation Lu=f with a zero Dirichlet boundary
condition is converted to an equivalent elliptic equation on the unit ball. A
spectral Galerkin method is applied to the reformulated problem, using
multivariate polynomials as the approximants. For a smooth boundary and smooth
problem parameter functions, the method is proven to converge faster than any
power of 1/n with n the degree of the approximate Galerkin solution. Examples
in two and three variables are given as numerical illustrations. Empirically,
the condition number of the associated linear system increases like O(N), with
N the order of the linear system.Comment: This is latex with the standard article style, produced using
Scientific Workplace in a portable format. The paper is 22 pages in length
with 8 figure
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Stability and covergence properties of Bergman Kernel methods for numerical conformal mapping
In this paper we study the stability and convergence properties of Bergman kernel methods, for the numerical conform al mapping of simply and doubly- connected domains. In particular, by using certain well-known 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
Numerical experiments with Bergman kernel functions in 2 and 3 dimensional cases
Pub. Int. CMAT, 1 (2003)In this paper we revisit the so-called Bergman kernel method - BKM- for solving conformal mapping problems and propose a generalized BKM-approach to extend the theory to 3-dimensional mapping problems. A special software package for quaternions was developed for the numerical experiments
A 3-dimensional Bergman Kernel method with applications to rectangular domains
In this paper we revisit the so-called Bergman kernel method - BKM - for solving conformal
mapping problems and propose a generalized BKM-approach to extend the theory to
3-dimensional mapping problems. A special software package for quaternions was developed
for the numerical experiments.Fundação para a Ciência e a Tecnologia (FCT
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