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\(L^{2}\)-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 (Mg) 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.

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Hardering, H. \(L^{2}\)-discretization error bounds for maps into Riemannian manifolds. Numer. Math. 139, 381–410 (2018). https://doi.org/10.1007/s00211-017-0941-3

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  • DOI: https://doi.org/10.1007/s00211-017-0941-3

Mathematics Subject Classification

  • 65N15
  • 65N30