Abstract:

Tullio Regge proposed in 1961 a novel approach for solving Einstein field equations in general relativity in a coordinate free fashion. This so-called Regge calculus was given a finite element context in the last 10 years by using symmetric and solely tangential-tangential continuous matrix-valued finite element functions to approximate metric and strain fields. In discrete differential geometry objects defined in the context of smooth manifolds are required to be computed on non-smooth structures and the natural question of approximation properties arises, when the discrete structure converges to its smooth counterpart.

In this talk we give an introduction into Regge finite elements. By approximating the metric tensor of a smooth Riemannian manifold we present and analyze procedures to compute geodesics and curvature quantities based on the discrete Regge finite elements. We focus on the scalar curvature in any dimensions and give an outlook approximating the full Riemann curvature tensor in arbitrary dimensions. Several numerical examples, implemented in the open-source finite element software NGSolve (www.ngsolve.org), supplement and confirm the analytic results.

This is joint work with Evan Gawlik (University of Hawaiʻi at Mānoa), Jay Gopalakrishnan (Portland State University), Joachim Schöberl (TU Wien), and Max Wardetzky (University of Göttingen).

This event takes place in hybrid form (in person and online on Zoom). Slides and additional materials are available on the Moodle service of the University of Vienna. If you want to participate, please write an email to matteo.tommasini@univie.ac.at