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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. 

The Whitney forms on a simplex T admit high-order generalizations that have received a great deal of attention in numerical analysis. Less well-known are the shadow forms of Brasselet, Goresky, and MacPherson. These forms generalize the Whitney forms, but have rational coefficients, allowing singularities near the faces of T. Motivated by numerical problems that exhibit these kinds of singularities, we introduce degrees of freedom for the shadow k-forms that are well-suited for finite element implementations. In particular, we show that the degrees of freedom for the shadow forms are given by integration over the k-dimensional faces of the blow-up T̃ of the simplex T. Consequently, we obtain an isomorphism between the cohomology of the complex of shadow forms and the cellular cohomology of T̃, which vanishes except in degree zero. Additionally, we discover a surprising probabilistic interpretation of shadow forms in terms of Poisson processes. This perspective simplifies several proofs and gives a way of computing bases for the shadow forms using a straightforward combinatorial calculation. 

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.