Discretization and convergence for harmonic maps into trees (Q1884733)

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.