Distortion estimates for barycentric coordinates on Riemannian simplices
Provides theoretical foundations for finite element methods on Riemannian manifolds, enabling rigorous error analysis for numerical simulations in geometry and physics.
The authors define barycentric coordinates on Riemannian simplices via Karcher's center of mass and show that the Riemannian metric differs from the Euclidean metric induced by edge lengths by an error that shrinks quadratically with the maximum edge length, enabling convergence results for finite element approximations on manifolds.
We define barycentric coordinates on a Riemannian manifold using Karcher's center of mass technique applied to point masses for n+1 sufficiently close points, determining an n-dimensional Riemannian simplex defined as a "Karcher simplex." Specifically, a set of weights is mapped to the Riemannian center of mass for the corresponding point measures on the manifold with the given weights. If the points lie sufficiently close and in general position, this map is smooth and injective, giving a coordinate chart. We are then able to compute first and second derivative estimates of the coordinate chart. These estimates allow us to compare the Riemannian metric with the Euclidean metric induced on a simplex with edge lengths determined by the distances between the points. We show that these metrics differ by an error that shrinks quadratically with the maximum edge length. With such estimates, one can deduce convergence results for finite element approximations of problems on Riemannian manifolds.