L²-Discretization Error Bounds for Maps into Riemannian Manifolds

2016/12/19 by Hanne Hardering, Hardering, Hanne
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Composite Material Mechanics #FOS: Mathematics #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.1612.06086

openalex publication_date 2016/12/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the approximation of functions that map a Euclidean domain Ω⊂ ℝ^d into an n-dimensional Riemannian manifold (M,g) minimizing an elliptic, semilinear energy in a function set H⊂ W^1,2(Ω,M). The approximation is given by a restriction of the energy minimization problem to a family of conforming finite-dimensional approximations S_h⊂ H. We provide a set of conditions on S_h such that we can prove a priori W^1,2- and 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. 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.

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