Volume 39, pp. 202-230, 2012.

Creating domain mappings

Kendall Atkinson and Olaf Hansen

Abstract

Consider being given a mapping $\varphi:S^{d-1}% \overset{1-1}{\underset{onto}{\longrightarrow}}\partial\Omega$, with $\partial\Omega$ the $\left( d-1\right) $-dimensional smooth boundary surface for a bounded open simply-connected region $\Omega$ in $\mathbb{R}% ^{d}$, $d\geq2$. We consider the problem of constructing an extension $\Phi:\overline{B}_{d}\overset{1-1}{\underset{onto}{\longrightarrow}}% \overline{\Omega}$ 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 $\Phi$ 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

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