Session I.5 - Geometric Integration and Computational Mechanics
Wednesday, June 14, 14:30 ~ 15:00
Curvature approximation in arbitrary dimension with Regge finite elements
Evan Gawlik
University of Hawaii, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
We study the curvature of Regge metrics on simplicial triangulations of dimension $N$. Here, a Riemannian metric is called a Regge metric if it is piecewise smooth and its tangential-tangential components are single-valued on every codimension-1 simplex in the triangulation. When such a metric is piecewise polynomial, it belongs to a finite element space called the Regge finite element space. Regge metrics are not classically differentiable, but it turns out that one can still make sense of their curvature in a distributional sense. In the lowest-order setting, the distributional curvature of a Regge metric is a linear combination of Dirac delta distributions supported on codimension-2 simplices $S$, weighted by the angle at $S$: $2\pi$ minus the sum of the dihedral angles incident at $S$. For piecewise polynomial Regge metrics of higher degree, the distributional curvature includes additional contributions involving the curvature in the interior of each $N$-simplex and the jump in the mean curvature across each codimension-1 simplex.
We study the convergence of the distributional curvature under refinement of the triangulation. We show that in the $H^{-2}$-norm, this convergence takes place at a rate of $O(h^{r+1})$ when a smooth Riemannian metric is interpolated by a piecewise polynomial Regge metric of degree $r \ge 0$ on a triangulation whose maximum simplex diameter is $h$, provided that either $N=2$ or $r \ge 1$.
Joint work with Yakov Berchenko-Kogan (Florida Institute of Technology, USA) and Michael Neunteufel (TU Wien, Austria).