$L^2$-Discretization Error Bounds for Maps into Riemannian Manifolds

We study the approximation of functions that map a Euclidean domain $\mathrm{\Omega }\subset {\mathbb{R}}^d$ into an $n$-dimensional Riemannian manifold $(M,g)$ minimizing an elliptic, semilinear energy in a function set $H\subset W^{1,2}(\mathrm{\Omega },M)$. The approximation is given by a restriction of the energy minimization problem to a family of conforming finite-dimensional approximations $S_h\subset H$. We provide a set of conditions on $S_h$ such that we can prove a priori $W^{1,2}$- and $L^2$-approximation error estimates comparable to standard Euclidean finite elements. This is done in an intrinsic framework, independently of embeddings of the manifold or the choice of coordinates. A special construction of approximations —geodesic finite elements— is shown to fulfill the conditions, and in the process extended to maps into the tangential bundle.