Generalizing Riemann curvature to Regge metrics (upcoming)
Date:
The slides are WIP and will be shared when finished.
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. Regge calculus was originally developed to solve Einstein field equations in general relativity by discretizing the metric tensor by piece-wise constant metrics and approximating the curvature using angle deficits.
Regge finite elements consist of piece-wise symmetric matrix-valued polynomials with single-valued tangential-tangential components. Due to their weak continuity, taking derivatives leads to distributions. Recently, a concept of approximating and analyzing curvature quantities like the Gauss curvature, scalar curvature, and Einstein tensor has been successfully developed.
In this talk, we discuss the definition of the distributional Riemann curvature tensor in any dimension [J. Gopalakrishnan, M. Neunteufel, J. Schöberl and M. Wardetzky, Generalizing Riemann curvature to Regge metrics, arXiv:2311.01603, 2024]. We prove that in a negative Sobolev-norm we obtain convergence towards the smooth curvature of rate k+1 if the discrete metrics interpolate the smooth metric tensor g into Regge finite elements of polynomial order k. In dimension N=2, the rate holds for all non-negative integers k, whereas for N>2, one requires k>0.
We confirm with numerical examples, implemented in the open-source finite element software NGSolve (www.ngsolve.org), that the theoretical rates are sharp.