Past Event: Oden Institute Seminar

High order approximations of curvature, curl and incompatibility on Riemannian manifolds

Prof. Jay Gopalakrishnan, Portland University

Abstract

Since many techniques in geometry and physics involve components of curvature, it is natural to ask how one can obtain high order curvature approximations. In two-dimensional Riemannian manifolds, intrinsic curvature is captured by the Gauss curvature, which can classically be computed solely using the manifold's metric.  However, in applications, we often only know the metric approximately, e.g., the metric may be obtained by numerically solving a partial differential equation using finite elements. When such a metric approximation is only piecewise smooth, classical formulas for computing the Gauss curvature cannot be applied. The goal of this talk is to present the correct generalization of classical formulas and explain how their correctness can be proved by numerical analysis.  Specifically, we show that when the metric tensor is given in the so-called Regge finite element space (which contains nonsmooth functions), then certain curvature approximations not only converge, but in fact superconverge. Similar statements can be proved also for other covariant operators on the manifold like curl, incompatibility, and the Levi-Civita connection. The intuition for our analysis will be provided by drawing parallels with simpler results for linear operators on Euclidean domains, known from finite element exterior calculus.  This is joint work with M. Neunteufel, J. Schöberl, and M. Wardetzky.

Biography

Jay Gopalakrishnan is a computational mathematician whose research centers around the design of accurate and efficient numerical methods for partial differential equations.  He has co-authored over ninety publications, including the first papers developing what are now known as HDG methods and DPG methods. He has worked at Bell Labs, University of Minnesota, Medtronic Inc, and for over a decade, at University of Florida. He currently holds an endowed chair at Portland State University in Oregon.