Numerical analysis of curvatures from Regge finite elements

Riemannian manifolds are manifolds that are not embedded in higher-dimensional spaces (the surface of the earth for example is embedded in three-dimensional space). However, by an additionally given metric tensor, one can measure lengths and angles (such as measuring lengths and angles on the earth`s ground). To measure the curvature of the manifold, which looks locally flat (the earth`s surface is curved and not a disc, even if we locally often assume it to be flat), we need the second derivative of the metric. Suppose this metric, however, is only given approximatively on a triangulation of the manifold. In that case, the metric is not smooth enough to take derivatives in the classical sense, only in the sense of distributions. Discrete differential geometry methods, such as measuring angles around a point, have been developed to compute curvatures. Analysis and extension to higher precision is, however, complicated in this setting. In this project, we aim to develop fundamental theoretical and numerical tools to connect and understand discrete differential geometry together with finite element methods (FEM). Therefore, we assume the approximate metric is represented by a so-called Regge finite element, which now describes a non-smooth Riemannian manifold. To define derivatives and distributions on this abstract manifold, we need to derive the space of smooth (test-)functions with respect to the Regge metric. Further, the proof of density results of function spaces is crucial. Afterwards, if the approximate metric approaches the exact one, the convergence will be investigated. This would build a fundamental basis for future definitions and analyses of partial differential equations on (approximated) Riemannian manifolds. The so-called Riemann curvature tensor contains all information about the curvatures of a Riemannian manifold. This project aims to approximate it using appropriate finite elements that mimic the intrinsic properties correctly. Additionally, we search for finite elements for the Ricci and Einstein tensor in arbitrary space dimensions. After successfully identifying them, we will analyse their convergence properties analytically and numerically. In this project, we will develop the finite element software NGDiffGeo to perform numerical simulations on (approximated) Riemannian manifolds efficiently. A suitable definition of classes will simplify and accelerate the treatment of differential geometry objects, which are coordinate- intensive and, therefore, error-prone. Further, we will implement efficient methods to approximate curvature tensors.