任意维数量曲率的有限元逼近
Finite element approximation of scalar curvature in arbitrary dimension
摘要：
我们分析了数量曲率在$N \ge 2$维的有限元离散。我们的分析重点是分段多项式插值的光滑黎曼度量$g$上的一个单纯三角形的多面体域$Ω\subset \mathbb{R}^N$具有最大元素直径$h$。我们表明，如果这样的插值$g_h$有多项式次数$r \ge 0$和具有单值切线-切向分量的余维-1单纯形，那么它承认一个自然的概念（致密化）数量曲率，收敛在$H^{-2}(Ω)$ -范数的（致密化）数量曲率的$g$在$O(h^{r+1})$的速度为$h \to 0$，只要无论是$N = 2$或$r \ge 1$。作为一种特殊情况，我们的结果意味着收敛在$H^{-2}(Ω)$的广泛使用的“角度亏损”近似高斯曲率的二维三角剖分，没有严格的假设上的内插度量$g_h$。我们提出的数值实验表明，我们的分析估计是尖锐的。
英文摘要：
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 $Ω\subset \mathbb{R}^N$ 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 $H^{-2}(Ω)$-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 $H^{-2}(Ω)$ 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.
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