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.
翻译:我们分析维度为$\ge 2美元 的缩度缩度的有限元素离散值。 我们的分析侧重于平滑的里曼尼里平滑的里曼尼里平滑的多角度中间线的片状多角度内分解器元 $\ Omega\ subset\ mathbb{R ⁇ N$, 最大元素直径为$h美元。 我们显示,如果这种中间值$g_h$( 美元+1美元) 具有多元度 $r\ g 美元, 并在 Codimenion-1 simplicles上拥有单一价值的正价- 切度构件, 然后它承认一种自然的( 斜度) 弧度曲度概念, 在 $% 2 (\ Omega) ( ) ( omega) 调和( localal) 曲线曲度的调和( leg- groupalalalation $) (美元), 这个特殊例子意味着我们目前使用的直径 的直径直径的直径的直径的直径分析结果。