Discrete solutions of Dirichlet problems, finite volumes, and circle packings
type: Item
instance of: scholarly article
title: Discrete solutions of Dirichlet problems, finite volumes, and circle packings
zbMATH Open document ID: 0945.30004
DOI: 10.1007/PL00009447
published in: Discrete \& Computational Geometry
publication date: 2000-10-08T00:00:00Z
review text: Convergence results are obtained for discrete solutions of Dirichlet problems for Poisson equations in the plane. The solutions are constructed for triangular grids, using the finite volume method where the volumes have sides perpendicular to, but not necessarily bisecting corresponding edges in underlying triangulations. It is shown that the discrete solutions of Poisson equations satisfy the maximum principle. Convergence of discrete solutions to the corresponding classical solutions is proved in the \(H^1\) norm and the \(L^2\) norms, under mild restrictions on the triangulations and volumes. With further restriction on the domain and the triangulations, uniform convergence on compacta is established. It is then shown that triangulations and volumes induced by circle packings have the desired properties, if the packings satisfy some natural combinatorial restrictions. The packings are then used via the discrete Riemann mapping theorem to pull the Dirichlet problem back to a problem in the disk. Finally, the connection between random walks induced by circle packings and Brownian motion in the plane is discussed.
reviewed by: F. D. Lesley
Mathematics Subject Classification ID: 30C30
zbMATH DE Number: 1309665
zbMATH Keywords: discrete Dirichlet problem
zbMATH Keywords: circle packings
zbMATH Keywords: Brownian motion
author: Tomasz Dubejko
MaRDI profile type: Publication
full work available at URL: pl00009447
OpenAlex ID: W2058943247
description: scientific article; zbMATH DE number 1309665