Finite element approximation of scalar curvature in arbitrary dimension

Abstract

We analyze finite element discretizations of scalar curvature in dimension N ≥ 2 N \ge 2 . Our analysis focuses on piecewise polynomial interpolants of a smooth Riemannian metric g g on a simplicial triangulation of a polyhedral domain Ω ⊂ R N \Omega \subset \mathbb {R}^N having maximum element diameter h h . We show that if such an interpolant g h g_h has polynomial degree r ≥ 0 r \ge 0 and possesses single-valued tangential-tangential components on codimension-1 simplices, then it admits a natural notion of (densitized) scalar curvature that converges in the H − 2 ( Ω ) H^{-2}(\Omega ) -norm to the (densitized) scalar curvature of g g at a rate of O ( h r + 1 ) O(h^{r+1}) as h → 0 h \to 0 , provided that either N = 2 N = 2 or r ≥ 1 r \ge 1 . As a special case, our result implies the convergence in H − 2 ( Ω ) H^{-2}(\Omega ) of the widely used “angle defect” approximation of Gaussian curvature on two-dimensional triangulations, without stringent assumptions on the interpolated metric g h g_h . We present numerical experiments that indicate that our analytical estimates are sharp.