  We construct and analyze finite element approximations of the Einstein tensor
in dimension $N \ge 3$. We focus on the setting where a smooth Riemannian
metric tensor $g$ on a polyhedral domain $\Omega \subset \mathbb{R}^N$ has been
approximated by a piecewise polynomial metric $g_h$ on a simplicial
triangulation $\mathcal{T}$ of $\Omega$ having maximum element diameter $h$. We
assume that $g_h$ possesses single-valued tangential-tangential components on
every codimension-1 simplex in $\mathcal{T}$. Such a metric is not classically
differentiable in general, but it turns out that one can still attribute
meaning to its Einstein curvature in a distributional sense. We study the
convergence of the distributional Einstein curvature of $g_h$ to the Einstein
curvature of $g$ under refinement of the triangulation. We show that in the
$H^{-2}(\Omega)$-norm, this convergence takes place at a rate of $O(h^{r+1})$
when $g_h$ is an optimal-order interpolant of $g$ that is piecewise polynomial
of degree $r \ge 1$. We provide numerical evidence to support this claim.

Numerical Analysis;Numerical Analysis;Differential Geometry