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In this paper, we present an algorithm for the computation of harmonic maps,and respectively, of the harmonic map heat flow between two closed Riemannianmanifolds. Our approach is based on the totally geodesic embedding of thetarget manifold into $\mathbb{R}^N$ . Since embeddings of Riemannian manifoldsinto Euclidean spaces can easily be made totally geodesic by extending theRiemannian metric in a certain way into some tubular neighbourhood, the herepresented approach is quite general. Totally geodesic embeddings allow toreformulate the harmonic map heat flow in a neighbourhood of the embeddedtarget manifold. This reformulation has the advantage that the problem becomesunconstrained: Instead of assuming a priori that the solution to the flow mapsinto the target manifold this fact becomes a property of the solution to theextended flow for special initial data. The solution space to the reformulatedproblem therefore exists of maps which are also allowed to map into the ambientspace of the target manifold. This simplifies the discretization of theproblem. Based on this observation, we here propose algorithms for thecomputation of the harmonic map heat flow and of harmonic maps. In contrast toprevious schemes, our algorithm does not make use of projections onto thetarget manifold, discrete tangential deformations, geodesic finite elements orof Lagrange multipliers. We prove error estimates in the stationary case andpresent some numerical tests at the end of the paper.

On the computation of harmonic maps by unconstrained algorithms based on totally geodesic embeddings