Curvature approximation with Finite Elements
Date:
The slides can be found here.
Abstract:
Approximating the curvature of embedded manifolds, such as surfaces in R3, or (pseudo-) Riemannian manifolds, for instance, 4D space-time in general relativity, is an important field in discrete differential geometry and numerical relativity. In the last few years, finite element methods have been developed to embed differential geometry in the context of finite elements, enabling rigorous numerical analysis. In this talk, we will present how to compute the Gauss curvature on a 2D manifold discretized by a triangulation, where so-called Regge finite elements approximate the metric. We motivate the arising terms and present theoretical and numerical convergence results. Then, an extension to the Riemann curvature tensor in any dimension is briefly outlined.