Geodesic finite elements on simplicial grids

SUMMARY We introduce geodesic finite elements as a conforming way to discretize partial differential equations for functions v : Ω →  M , where Ω is an open subset ofR dand M is a Riemannian manifold. These geodesic finite elements naturally generalize standard first‐order finite elements for Euclidean spaces. They also generalize the geodesic finite elements proposed for d  = 1 in a previous publication of the author. Our formulation is equivariant under isometries of M and, hence, preserves objectivity of continuous problem formulations. We concentrate on partial differential equations that can be formulated as minimization problems. Discretization leads to algebraic minimization problems on product manifolds M n . These can be solved efficiently using a Riemannian trust‐region method. We propose a monotone multigrid method to solve the constrained inner problems with linear multigrid speed. As an example, we numerically compute harmonic maps from a domain inR 3to S 2 . Copyright © 2012 John Wiley & Sons, Ltd.