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 $\mathrm{\Omega }\subset {\mathbb{R}}^2$ with boundary $\partial \mathrm{\Omega }$ that is given by a smooth function $\phi :[a,b]\mapsto {\mathbb{R}}^2.$ Our goal is to calculate a polynomial $P^{(n)}:{\mathbb{B}}^2\mapsto \mathrm{\Omega }$ 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 $\mathrm{\Phi }$ that is only known for a discrete set of points in ${\mathbb{B}}^2$. The construction of $\mathrm{\Phi }$ 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 $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$.

Full Text (PDF) [981 KB], BibTeX , DOI: 10.1553/etna_vol60s351

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