Numerical Approximation in Riemannian Manifolds by Karcher Means
For researchers in computational geometry and finite element methods on manifolds, this provides rigorous error bounds for approximating Riemannian structures by piecewise flat simplicial complexes.
The paper establishes approximation estimates for Riemannian metrics and connections on Karcher simplices, showing that the error is bounded by O(h^2) for the metric and O(h) for the connection, assuming small edge lengths and nondegenerate simplex volumes. It also provides a cochain map from simplicial cohomology to a space of non-conforming differential forms, enabling variational problem reduction.
(1) For a compact Riemannian manifold without boundary $(M,g)$ containing $n+1$ points $p_i$ and the $n$-dimensional standard simplex $Δ$, the miniser of \[ E: M \times Δ\to {\mathbf R}, (a,λ) \mapsto λ^0 d^2(a,p_0) + \dots + λ^n d^2(a,p_n) \] is considered as point with "barycentric coordinates" $λ_i$ 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 $Δ$ 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 $vol(Δ,g^e) \geq αh^n$ for some $α> 0$, 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 $α$. 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 $P^{-1}Ω^k \subset L^\inftyΩ^k$ and outer differentials and give an isometric cochain map $C^k \to P^{-1}Ω^k$. 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 $P^{-1}Ω^k$ in $H^1Ω^k$ and compare the solutions to variational problems in both spaces.