Finite element approximation of the Levi-Civita connection and its curvature in two dimensions

TL;DR

This paper develops finite element methods using Regge finite elements to approximate the Levi-Civita connection and curvature on 2D manifolds, ensuring convergence in a distributional sense.

Contribution

It introduces a novel finite element approach for discretizing connections and curvature in 2D, handling low regularity via distributional formulations and proving convergence.

Findings

Distributional curvature and connection converge to smooth counterparts with mesh refinement

Projection operators commute with linearized differential operators, preserving complex structures

Finite element approximations are valid in a distributional framework for low-regularity metrics

Abstract

We construct finite element approximations of the Levi-Civita connection and its curvature on triangulations of oriented two-dimensional manifolds. Our construction relies on the Regge finite elements, which are piecewise polynomial symmetric (0,2)-tensor fields possessing single-valued tangential-tangential components along element interfaces. When used to discretize the Riemannian metric tensor, these piecewise polynomial tensor fields do not possess enough regularity to define connections and curvature in the classical sense, but we show how to make sense of these quantities in a distributional sense. We then show that these distributional quantities converge in certain dual Sobolev norms to their smooth counterparts under refinement of the triangulation. We also discuss projections of the distributional curvature and distributional connection onto piecewise polynomial finite element spaces. We show that the relevant projection operators commute with certain linearized differential operators, yielding a commutative diagram of differential complexes.