Distributional computation of intrinsic curvature with Regge finite elements
Date:
The slides can be found here.
Abstract:
The intrinsic curvature of a metric tensor defined in the parameter domain \(\Omega\subset\mathbb{R}^d\) is given by the (Riemann) curvature tensor \(Q\). Regge calculus has been originally developed for solving problems in general relativity by approximating the curvature tensor by piece-wise constant metrics. Later, a finite element context was given leading to arbitrary polynomial order, tangential-tangential continuous, and symmetric so-called Regge elements discretizing metric tensors.
In this talk we present a novel arbitrary order intrinsic curvature computation procedure based on a lifting of the nonlinear distributional curvature by means of Regge elements. For two-dimensional surfaces we make use of the Gauss–Bonnet theorem leading to the distributional vertex and edge terms. An extension to three-dimensional sub-manifolds in four dimensions is discussed. Numerical experiments, implemented in the finite element software NGSolve (www.ngsolve.org), are presented validating the optimal convergence rates of the proposed method.