Analysis of distributional Riemann curvature tensor in any dimension
Date:
The slides can be found here.
Abstract:
The intrinsic curvature of an \(N\)-dimensional Riemannian manifold \((M,g)\) with metric tensor \(g\) is given by the fourth order Riemann curvature tensor \(\mathcal{R}(g)\). Regge calculus \cite{Neunteufel:Regge} has originally been developed for solving Einstein field equations in general relativity by discretizing the metric tensor by piece-wise constant metrics and approximating the curvature by means of angle deficits.
Regge finite elements have been recently developed [S.H. Christiansen, On the linearization of Regge calculus, Numerische Mathematik 119, 4 (2011)], [L. Li, Regge Finite Elements with Applications in Solid Mechanics and Relativity, PhD thesis, University of Minnesota (2018)] consisting of piece-wise symmetric matrix-valued polynomials with weak continuity over element interfaces. Only the tangential-tangential components are single-valued. Theses elements turned out to have beneficial properties for discretizing strain and metric fields in several applications as continuum mechanics, shells, and differential geometry.
Due to the weak continuity of Regge elements taking derivatives lead to distributions. In the last 4 years a concept of approximating and analyzing curvature quantities like the Gauss curvature and connection 1-form in 2D [Y. Berchenko-Kogan and E.S. Gawlik, Finite element approximation of the Levi-Civita connection and its curvature in two dimensions, Foundations of Computational Mathematics (2022)], [J. Gopalakrishnan, M. Neunteufel, J. Schöberl, M. Wardetzky, Analysis of curvature approximations via covariant curl and incompatibility for Regge metrics, SMAI J. Comput. Math. 9 (2023)] and scalar curvature in any dimension [E.S. Gawlik, M. Neunteufel, Finite element approximation of scalar curvature in arbitrary dimension, arXiv, 2301.02159 (2023)] have been successfully developed.
In this talk we propose a novel definition of the distributional (denisitized) Riemann curvature tensor \(\widetilde{\mathcal{R}\omega}(g_h)\) in any dimension [J. Gopalakrishnan, M. Neunteufel, J. Schöberl, M. Wardetzky, Analysis of distributional Riemann curvature tensor in any dimension. (in preparation)]. We prove that in the \(H^{-2}\)-norm we obtain convergence towards the smooth curvature \((\mathcal{R}\omega)(g)\) of rate \(\mathcal{O}(h^{k+1})\) if \(g_h\) is a sequence of Regge metrics interpolating the smooth metric tensor \(g\) into Regge finite elements of polynomial order \(k\). In dimension \(N=2\) the rate holds for all \(k\geq 0\), whereas for $N\geq 3$ one requires at least linear elements \(k\geq 1\).
We confirm with numerical examples, implemented in the open source finite element software NGSolve (www.ngsolve.org), that the theoretical rates are sharp.