Numerical Approximation in Riemannian Manifolds by Karcher Means

Abstract: (1) For a compact Riemannian manifold without boundary $(M,g)$ containing $n+1$ points $p_i$ and the $n$-dimensional standard simplex $\Delta$, the miniser of [ E: M \times \Delta \to {\mathbf R}, (a,\lambda) \mapsto \lambda^0 d^2(a,p_0) + \dots + \lambda^n d^2(a,p_n) ] is considered as point with "barycentric coordinates" $\lambda_i$ within the so-called Karcher simplex (or Riemannian simplex or geodesic finite element) defined by vertices $p_i$. In the small, existence and uniqueness is well-known. Now suppose $\Delta$ carries a flat Riemannian metric $g^e$ induced by edge lengths $d(p_i,p_j)$, where $d$ is the geodesic distance in $M$. If all edge lengths are small than $h$ and $vol(\Delta,g^e) \geq \alpha h^n$ for some $\alpha > 0$, then we can show that \begin{equation} |(x^*g - g^e)(v,w)| \leq c h^2 |v| |w|, \qquad |(\nabla^{x^*g} - \nabla^{g^e})_v w| \leq c h |v| |w| \end{equation} with some constant $c$ depending only on the curvature tensor $R$ of $(M,g)$ and $\alpha$. From this we derive several estimates for Finite Element calculations in which $(M,g)$ is replaced by a piecewise flat realised simplicial complex. (2) Let $M$ be the geometric realisation of a simplicial complex $K$. The simplicial cohomology $(C^k(K), \partial^*)$ has been interpreted as "discrete outer calculus" (DEC) in the literature. We define spaces $P^{-1}\Omega^k \subset L^\infty\Omega^k$ and outer differentials and give an isometric cochain map $C^k \to P^{-1}\Omega^k$. This reduces the computation of variational problems in discrete outer calculus to variational problems in a trial space of non-conforming differential forms. We investigate the approximation properties of $P^{-1}\Omega^k$ in $H^1\Omega^k$ and compare the solutions to variational problems in both spaces.