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.