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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.
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Finite element approximation of the Einstein tensor
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.
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- PAR ID:
- 10579675
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- IMA Journal of Numerical Analysis
- ISSN:
- 0272-4979
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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