Finite Element Approximation of the Levi-Civita Connection and Its Curvature in Two Dimensions

Abstract

The Whitney forms on a simplex T admit high-order generalizations that have received a great deal of attention in numerical analysis. Less well-known are the shadow forms of Brasselet, Goresky, and MacPherson. These forms generalize the Whitney forms, but have rational coefficients, allowing singularities near the faces of T . Motivated by numerical problems that exhibit these kinds of singularities, we introduce degrees of freedom for the shadow k-forms that are wellsuited for finite element implementations. In particular, we show that the degrees of freedom for the shadow forms are given by integration over the k-dimensional faces of the blow-up T of the simplex T . Consequently, we obtain an isomorphism between the cohomology of the complex of shadow forms and the cellular cohomology of T , which vanishes except in degree zero. Additionally, we discover a surprising probabilistic interpretation of shadow forms in terms of Poisson processes. This perspective simplifies several proofs and gives a way of computing bases for the shadow forms using a straightforward combinatorial calculation.

Finite element approximation of scalar curvature in arbitrary dimension

“…It can be regarded as the integral of div div Sσ against v, where div div is interpreted in a distributional sense. This link with the Hellan-Herrmann-Johnson method has previously been noted and used in dimension N = 2 [4,16,19].…”

“…3. Berchenko-Kogan and Gawlik [4,Corollary 6.2] proved that if r ≥ 1, N = 2, and g h is any optimal-order interpolant of g, then (Rω) dist (g h ) converges to (Rω)(g) at a rate of O(h r ) in the norm u V ′ ,h = sup v∈V,v =0 u, v V ′ ,V / v V,h , where…”

“…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.…”

“…Another key aspect of our analysis is our use of the H −2 (Ω)-norm to measure the error. This norm is weaker than the ones used in [4,16,19], and it appears to be more natural for measuring the error in the curvature. For example, for piecewise constant Regge metrics in dimension N = 2, we show that convergence of (Rω) dist (g h ) to (Rω)(g) holds in the H −2 (Ω)-norm for any optimalorder interpolant of g, but numerical experiments suggest that it fails to hold in stronger norms when g h is not the canonical interpolant of g. A key tool that we use to prove convergence in H −2 (Ω) is the near-equivalence of a certain pair of metric-dependent, mesh-dependent norms on V ; see Proposition 4.5.…”