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