Finite element approximation of scalar curvature in arbitrary dimension
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We analyze finite element discretizations of scalar curvature in dimension N ≥ 2. Our analysis focuses on piecewise polynomial interpolants of a smooth Riemannian metric g on a simplicial triangulation of a polyhedral domain Ω⊂ℝ^N having maximum element diameter h. We show that if such an interpolant g_h has polynomial degree r ≥ 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)-norm to the (densitized) scalar curvature of g at a rate of O(h^r+1) as h → 0, provided that either N = 2 or r ≥ 1. As a special case, our result implies the convergence in H^-2(Ω) 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.
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