365 IGPM365.pdf        May 2013
TITLE Optimal A Priori Discretization Error Bounds for Geodesic Finite Elements
AUTHORS Philipp Grohs, Hanne Hardering, Oliver Sander
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.
KEYWORDS geodesic finite elements, discretization error, a priori bounds
DOI 10.1007/s10208-014-9230-z
PUBLICATION Foundations of Computational Mathematics (FoCM)