## View abstract

### 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. document.getElementById('cloak226ca0540eb000bb89b2e5af8bca4089').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy226ca0540eb000bb89b2e5af8bca4089 = '&#101;g&#97;wl&#105;k' + '&#64;'; addy226ca0540eb000bb89b2e5af8bca4089 = addy226ca0540eb000bb89b2e5af8bca4089 + 'h&#97;w&#97;&#105;&#105;' + '&#46;' + '&#101;d&#117;'; var addy_text226ca0540eb000bb89b2e5af8bca4089 = '&#101;g&#97;wl&#105;k' + '&#64;' + 'h&#97;w&#97;&#105;&#105;' + '&#46;' + '&#101;d&#117;';document.getElementById('cloak226ca0540eb000bb89b2e5af8bca4089').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy226ca0540eb000bb89b2e5af8bca4089 + '\'>'+addy_text226ca0540eb000bb89b2e5af8bca4089+'<\/a>';

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).