Analysis of curvature approximations via covariant curl and incompatibility for Regge metrics

Abstract

The metric tensor of a Riemannian manifold can be approximated using Regge finite elements and such approximations can be used to compute approximations to the Gauss curvature and the Levi-Civita connection of the manifold. It is shown that certain Regge approximations yield curvature and connection approximations that converge at a higher rate than previously known. The analysis is based on covariant (distributional) curl and incompatibility operators which can be applied to piecewise smooth matrix fields whose tangential-tangential component is continuous across element interfaces. Using the properties of the canonical interpolant of the Regge space, we obtain superconvergence of approximations of these covariant operators. Numerical experiments further illustrate the results from the error analysis.

Finite element approximation of scalar curvature in arbitrary dimension

“…We prove a convergence result in the case where N = 2, r = 0, and g h is an arbitrary optimal-order interpolant of g. This has been a longstanding gap in the literature on Gaussian curvature approximation. Previous efforts to address the case where N = 2 and r = 0 have relied on subtle properties of the geodesic interpolant [8] and the canonical interpolant [19].…”

“…The approach above is similar to the one used in dimension N = 2 in [4,16,19], but there are a few important differences. First, we work with an integral formula for the error (Rω) dist (g h ) − (Rω)(g) rather than an integral formula for the curvature itself.…”

“…First, we work with an integral formula for the error (Rω) dist (g h ) − (Rω)(g) rather than an integral formula for the curvature itself. Previous analyses in [4,16,19] hinged on formulas of the latter type. Loosely speaking, in this paper we compute the evolution of the error along a one-parameter family of Regge metrics starting at g and ending at g h , whereas the papers [4,16,19] compute the evolution of the curvature along a pair of one-parameter families of metrics: one family that starts at the Euclidean metric δ and ends at g h , and one that starts at δ and ends at g. The approach based on evolving the error appears to be better suited for proving optimal error estimates.…”

Oberwolfach report : Discretization of Hilbert complexes

“…The situation can be more subtle for nonlinear problems since the product of Dirac measures is not defined in general (c.f. [16,32,43]).…”

“…General constructions of distributional and nonconforming elements for the BGG complexes call for further investigation. These elements enjoy simple degrees of freedom, and may thus provide a bridge for discretization of PDEs and discrete structures, including graph theory [61], discrete mechanics [48], discrete differential geometry [16,31,37,43,59] and gauge theory [34]. Finite elements may provide a new perspective for these areas by supplying local shape functions [31] and inspire new schemes.…”