Grohs, P. and Hardering, H. and Sander, O. (2015) Optimal a priori discretization error bounds for geodesic finite elements. Foundations of Computational Mathematics, 15 (6). pp. 13571411. ISSN 16153375

PDF
630kB 
Official URL: http://dx.doi.org/10.1007/s102080149230z
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 wellknown 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 manifoldvalued variational problems. We measure the discretization error both intrinsically using an H1type Finsler norm and with the H1norm using embeddings of the codomain in a linear space. To measure the regularity of the solution, we propose a nonstandard smoothness descriptor for manifoldvalued 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 highorder scheme for this problem.
Item Type:  Article 

Uncontrolled Keywords:  Geodesic finite elements, a priori error estimates, harmonic maps, highorder methods 
Subjects:  Mathematical and Computer Sciences > Mathematics > Numerical Analysis 
Divisions:  Department of Mathematics and Computer Science > Institute of Mathematics 
ID Code:  1821 
Deposited By:  Ekaterina Engel 
Deposited On:  23 Feb 2016 09:29 
Last Modified:  03 Mar 2017 14:42 
Repository Staff Only: item control page