Volume 39, pp. 202-230, 2012.

Creating domain mappings

Kendall Atkinson and Olaf Hansen

Abstract

Consider being given a mapping $\phi :S^{d-1}\stackrel{1-1}{\underset{onto}{⟶}}\partial \mathrm{\Omega }$, with $\partial \mathrm{\Omega }$ the $(d-1)$-dimensional smooth boundary surface for a bounded open simply-connected region $\mathrm{\Omega }$ in ${\mathbb{R}}^d$, $d\ge 2$. We consider the problem of constructing an extension $\mathrm{\Phi }:{\bar{B}}_d\stackrel{1-1}{\underset{onto}{⟶}}\bar{\mathrm{\Omega }}$ with $B_d$ the open unit ball in ${\mathbb{R}}^d$. The mapping is also required to be continuously differentiable with a non-singular Jacobian matrix at all points. We discuss ways of obtaining initial guesses for such a mapping $\mathrm{\Phi }$ and of then improving it by an iteration method.

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Key words

domain mapping, multivariate polynomial, constrained minimization, nonlinear iteration

AMS subject classifications

65D99

Links to the cited ETNA articles

[2]	Vol. 17 (2004), pp. 206-217 Kendall Atkinson and Weimin Han: On the numerical solution of some semilinear elliptic problems
[4]	Vol. 37 (2010), pp. 386-412 Kendall Atkinson and Olaf Hansen: A spectral method for the eigenvalue problem for elliptic equations

ETNA articles which cite this article

Vol. 55 (2022), pp. 671-686 Kendall Atkinson, David Chien, and Olaf Hansen: Constructing diffeomorphisms between simply connected plane domains

Vol. 60 (2024), pp. 351-363 Kendall Atkinson, David Chien, and Olaf Hansen: Constructing diffeomorphisms between simply connected plane domains-part 2