$$L^{2}$$L2-discretization error bounds for maps into Riemannian manifolds
Abstract
We study the approximation of functions that map a Euclidean domain $$\Omega \subset {\mathbb {R}}^{d}$$ΩźRd into an n-dimensional Riemannian manifold (M, g) minimizing an elliptic, semilinear energy in a function set $$H\subset W^{1,2}(\Omega,M)$$HźW1,2(Ω,M). The approximation is given by a restriction of the energy minimization problem to a family of conforming finite-dimensional approximations $$S_{h}\subset H$$ShźH. We provide a set of conditions on $$S_{h}$$Sh such that we can prove a priori $$W^{1,2}$$W1,2- and $$L^{2}$$L2-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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- $$L^{2}$$L2-discretization error bounds for maps into Riemannian manifolds
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Published: 01 June 2018
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