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Optimal A Priori Discretization Error Bounds for Geodesic Finite Elements

by P. Grohs and H. Hardering and O. Sander

(Report number 2013-16)

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 \(H^1\)-type Finsler norm, and with the \(H^1\)-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.

Keywords: Geodesic finite elements, Riemannian center of mass, geometric PDEs, C\'ea Lemma, Approximation Error

@Techreport{GHS13_512,
  author = {P. Grohs and H. Hardering and O. Sander},
  title = {Optimal A Priori Discretization Error Bounds for Geodesic Finite Elements},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2013-16},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2013/2013-16.pdf },
  year = {2013}
}

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