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Optimal A Priori Discretization Error Bounds for Geodesic Finite Elements

Grohs, Philipp ; Hardering, Hanne ; Sander, Oliver

In: Foundations of Computational Mathematics, 2015, vol. 15, no. 6, p. 1357-1411

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    Title
    • Optimal A Priori Discretization Error Bounds for Geodesic Finite Elements
    Author
    • Grohs, Philipp. Zurich, Switzerland
    • Hardering, Hanne. Zurich, Switzerland
    • Sander, Oliver. Zurich, Switzerland
    Document Type
    • Postprint
    Language
    • English
    Published in
    • Foundations of Computational Mathematics, 2015, vol. 15, no. 6, p. 1357-1411. Springer US; http://www.springer-ny.com
    Other Version
    OAI-PMH Identifier
    • oai:doc.rero.ch:331261
    Summary
    We prove optimal bounds for the discretization error of geodesic finite elements for variational partial differential equations for functions that map into a nonlinear space. For this, we first generalize the well-known Céa lemma to nonlinear function spaces. In a second step, we prove optimal interpolation error estimates for pointwise interpolation by geodesic finite elements of arbitrary order. These two results are both of independent interest. Together they yield optimal a priori error estimates for a large class of manifold-valued variational problems. We measure the discretization error both intrinsically using an $$H^1$$ H 1 -type Finsler norm and with the $$H^1$$ H 1 -norm using embeddings of the codomain in a linear space. To measure the regularity of the solution, we propose a nonstandard smoothness descriptor for manifold-valued functions, which bounds additional terms not captured by Sobolev norms. As an application, we obtain optimal a priori error estimates for discretizations of smooth harmonic maps using geodesic finite elements, yielding the first high-order scheme for this problem.