Finite Element Approximation of Scalar Curvature in Arbitrary Dimension

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

Download Bibtex