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Abstract We construct and analyse 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 $$\varOmega \subset \mathbb{R}^{N}$$ has been approximated by a piecewise polynomial metric $$g_{h}$$ on a simplicial triangulation $$\mathcal{T}$$ of $$\varOmega $$ 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 $$H^{-2}(\varOmega )$$-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. In the process of proving our convergence results we derive a few formulas for the evolution of certain geometric quantities under deformations of the metric.