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：我们构造并分析维度 $N \ge 3$ 下爱因斯坦张量的有限元逼近。本文关注如下设定：多面体域 $\Omega \subset \mathbb{R}^N$ 上的光滑黎曼度量张量 $g$ 由 $\Omega$ 的单纯形三角剖分 $\mathcal{T}$ 上的分片多项式度量 $g_h$ 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。此类度量通常不具有经典可微性，但研究表明其爱因斯坦曲率仍可在分布意义下被赋予意义。我们研究在三角剖分细化过程中 $g_h$ 的分布爱因斯坦曲率向 $g$ 的爱因斯坦曲率的收敛性。结果表明，当 $g_h$ 为 $g$ 的最优阶分片 $r \ge 1$ 次多项式插值时，在 $H^{-2}(\Omega)$-范数下该收敛速度可达 $O(h^{r+1})$。我们提供数值实验验证该结论。