Discretization and convergence for harmonic maps into trees
type: Item
instance of: scholarly article
title: Discretization and convergence for harmonic maps into trees
zbMATH Open document ID: 1059.58008
author: Yanyan Li
published in: Calculus of Variations and Partial Differential Equations
publication date: 2004-11-05T00:00:00Z
review text: While originally intended for maps between Riemannian manifolds, harmonic maps have recently been extended to targets being metric spaces of non-positive curvature in the sense of Alexandrov (NPC spaces), for example trees. Following Jost, the authors consider the nonlinear Dirichlet problem for harmonic maps from a measure space equipped with a local Dirichlet form into trees with one branch point, called spiders. In such a setting the energy functional can still be defined, using the Markov kernels associated to the generator of the Dirichlet form on the domain. Specialising to the case of \({\mathbb R}^2\) and its classical Dirichlet form, so that the energy functional becomes the usual one, an algorithm is developed to solve the Dirichlet problem of finding harmonic maps on \({\mathbb R}^2\) with fixed values outside a domain chosen to be a polygonal \(D\). This relies on triangulations of \(D\) and a discrete version of the energy for mappings from the vertices, yielding a discrete nonlinear Dirichlet problem, solved algorithmically by iteration of Markov operators. Prolongation operators extend maps on the vertices to the whole of \(D\). Numerical results are presented.
reviewed by: Eric Loubeau
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 31C25
Mathematics Subject Classification ID: 60J10
zbMATH DE Number: 2113879
zbMATH Keywords: harmonic maps
zbMATH Keywords: Dirichlet problems
zbMATH Keywords: discretization
MaRDI profile type: Publication
full work available at URL: s00526-003-0248-3
OpenAlex ID: W2037414324
DOI: 10.1007/S00526-003-0248-3
description: scientific article; zbMATH DE number 2113879