|Preprint-No.:||< 365 >||Published in:||May 2013||PDF-File:||IGPM365_k.pdf|
|Title:||Optimal A Priori Discretization Error Bounds for Geodesic Finite Elements|
|Authors:||Philipp Grohs, Hanne Hardering, Oliver Sander|
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.
|Keywords:||geodesic finite elements, discretization error, a priori bounds|
|Publication:||Foundations of Computational Mathematics (FoCM)|