Numerical analysis of a finite element scheme for the approximation of harmonic maps into surfaces

This article studies the numerical approximation of harmonic maps into surfaces, i.e., critical points for the Dirichlet energy among weakly differentiable vector fields that are constrained to attain their pointwise values in a given manifold. An iterative algorithm that is based on a linearization of the constraint about the current iterate at the nodes of a triangulation is devised and its global convergence to a discrete harmonic map is proved under general conditions. Weak accumulation of discrete harmonic maps at harmonic maps as discretization parameters tend to zero is established in two dimensions under certain assumptions on the underlying sequence of triangulations. Numerical simulations illustrate the performance of the algorithm for curved domains.