Finite Element Approximation of the Einstein Tensor

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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