Optimal A Priori Discretization Error Bounds for Geodesic Finite Elements

Publikation: Beitrag in FachzeitschriftForschungsartikelBeigetragenBegutachtung

Beitragende

  • Philipp Grohs - , ETH Zurich (Autor:in)
  • Hanne Hardering - , Freie Universität (FU) Berlin (Autor:in)
  • Oliver Sander - , Rheinisch-Westfälische Technische Hochschule Aachen (Autor:in)

Abstract

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 𝐻1-type Finsler norm and with the 𝐻1-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.

Details

OriginalspracheEnglisch
Seiten (von - bis)1357-1411
Seitenumfang55
FachzeitschriftFoundations of Computational Mathematics
Jahrgang15
Ausgabenummer6
PublikationsstatusVeröffentlicht - 2015
Peer-Review-StatusJa
Extern publiziertJa

Externe IDs

Scopus 84945454765
ORCID /0000-0003-1093-6374/work/142250572

Schlagworte

Schlagwörter

  • geodesic finite elements, discretization error, interpolation error, optimal bounds