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.

翻译：我们针对维度$N \ge 3$的情形，构建并分析了爱因斯坦张量的有限元逼近。研究聚焦于如下设定：在多面体区域$\Omega \subset \mathbb{R}^N$上，光滑黎曼度量张量$g$被定义于$\Omega$的单纯形剖分$\mathcal{T}$上的分片多项式度量$g_h$所逼近，其中$\mathcal{T}$的最大单元直径为$h$。我们假设$g_h$在$\mathcal{T}$的每个余维数为1的单纯形上具有单值切向-切向分量。一般而言，此类度量在经典意义下不可微，但可对其爱因斯坦曲率赋予分布意义。我们研究了在网格细化过程中，$g_h$的分布意义爱因斯坦曲率向$g$的爱因斯坦曲率的收敛性。结果表明：当$g_h$是$g$的最优阶插值逼近，且为次数$r \ge 1$的分片多项式时，该收敛在$H^{-2}(\Omega)$范数下的收敛阶为$O(h^{r+1})$。我们提供了数值实验支持这一结论。