Volume 60, pp. 351-363, 2024.

Constructing diffeomorphisms between simply connected plane domains-part 2

Kendall Atkinson, David Chien, and Olaf Hansen

Abstract

Consider a simply connected domain $\Omega\subset\mathbb{R}^2$ with boundary $\partial\Omega$ that is given by a smooth function $\varphi: [a,b]\mapsto \mathbb{R}^2.$ Our goal is to calculate a polynomial $P^{(n)}:\mathbb{B}^{2}\mapsto \Omega$ of maximum degree $n$ such that $P^{(n)}$ is a diffeomorphism. Here $\mathbb{B}^{2}$ is the open unit disk in $\mathbb{R}^2$, and $n$ has to be chosen suitably large. The polynomial mapping $P^{(n)}$ is given as the $L^2$-projection of a mapping $\Phi$ that is only known for a discrete set of points in $\mathbb{B}^{2}$. The construction of $\Phi$ was given in a previous article of the authors [Electron., Trans., Numer., Anal., 55 (2022), pp. 671–686]. Using $P^{(n)}$ we can transform boundary value problems on $\Omega$ to analogous ones on $\mathbb{B}^{2}$ and then solve them using a Galerkin method. In Section 5 we give numerical examples demonstrating the use of $P^{(n)}$ to solve Dirichlet problems for two example regions $\Omega$.

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Key words

domain mapping, multivariate polynomial, constrained minimization, nonlinear iteration

AMS subject classifications

65D05, 49Q10

Links to the cited ETNA articles

[3]	Vol. 55 (2022), pp. 671-686 Kendall Atkinson, David Chien, and Olaf Hansen: Constructing diffeomorphisms between simply connected plane domains
[4]	Vol. 39 (2012), pp. 202-230 Kendall Atkinson and Olaf Hansen: Creating domain mappings

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