$$L^{2}$$L2-discretization error bounds for maps into Riemannian manifolds

Abstract

We study the approximation of functions that map a Euclidean domain \(\Omega \subset {\mathbb {R}}^{d}\) into an n-dimensional Riemannian manifold (M, g) minimizing an elliptic, semilinear energy in a function set \(H\subset W^{1,2}(\Omega ,M)\). 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.