Finite element approximation of scalar curvature in arbitrary dimension

TL;DR

This paper proves that finite element methods for approximating scalar curvature on triangulated domains converge at a specific rate, extending known results and including practical angle defect methods, with numerical validation.

Contribution

It establishes convergence rates for finite element scalar curvature approximations in arbitrary dimensions, including the angle defect method in 2D, under minimal assumptions.

Findings

Convergence in $H^{-2}$ norm at rate $O(h^{r+1})$ for polynomial degree $r \\ge 1$

Angle defect approximation converges without strict metric assumptions in 2D

Numerical experiments confirm the sharpness of the theoretical estimates.

Abstract

We analyze finite element discretizations of scalar curvature in dimension $N \ge 2$. 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 $r \ge 0$ 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 $H^{-2}(\Omega)$-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 $r \ge 1$. 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.