M. Hesse, M. Rumpf, and K.-T. Sturm.
Discretization and convergence for harmonic maps into trees.
Calculus of Variations, 21:113-136, 2004.
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The nonlinear Dirichlet problem is considered for maps from a two dimensional domain into trees with one branch point. The Dirichlet energy is defined using a semigroup approach based on Markov kernels. The problem is discretized using a suitable finite element approach and convergence of a corresponding iterative numerical method is proved. The presented approach integrates stochastic methods on discrete lattices and finite element projection techniques. Finally, a couple of numerical results are presented.