Analysis of intrinsic curvature approximations with Regge finite elements
Date:
The slides can be found here.
Abstract:
The intrinsic curvature of a Riemannian manifold \((M,g)\) is given by the curvature tensor \(Q\). Regge calculus [T. Regge, General relativity without coordinates. Il Nuovo Cimento (1955-1965) 19, 3 (1961), 558–571] has originally been developed for solving Einstein field equations in general relativity by discretizing the metric tensor \(g\) by piece-wise constant metrics and approximating the curvature \(Q\) by means of angle deficits. For two-dimensional manifolds, a proof of convergence for curvature using (high-order) Regge finite elements was recently given in [Y. Berchenko-Kogan and E.~S. Gawlik, Finite element approximation of the Levi-Civita connection and its curvature in two dimensions, arXiv preprint, 2111.02512 (2021)].
In this talk we present an improved error analysis obtaining one extra order of convergence and confirm with numerical examples, implemented in the finite element software NGSolve (www.ngsolve.org), that the rates are optimal. Further, an extension of the high-order curvature approximation in three dimensions is presented leading to optimal numerical convergence rates.