Finite element approximation of the Einstein tensor

TL;DR

This paper develops finite element methods to approximate the Einstein tensor for smooth Riemannian metrics on polyhedral domains, demonstrating convergence in a distributional sense with quantifiable rates.

Contribution

It introduces a finite element approach for approximating the Einstein tensor in a distributional framework, with proven convergence rates for piecewise polynomial metric interpolants.

Findings

Convergence of Einstein curvature approximation in the $H^{-2}$ norm at rate $O(h^{r+1})$.

Numerical experiments support theoretical convergence results.

Abstract

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.