Finite Element Systems for vector bundles : elasticity and curvature

TL;DR

This paper develops a comprehensive theory of Finite Element Systems for discretizing vector bundle sections, with applications to elasticity and curvature, including new finite element spaces and discrete geometric identities.

Contribution

It introduces a unified framework for finite element discretizations of vector bundles, proving discrete geometric identities and constructing new finite element spaces with curvature properties.

Findings

Proves a discrete Bianchi identity in curved settings

Establishes a de Rham theorem for flat cases

Defines new conforming finite element spaces in 2D for metrics with curvature

Abstract

We develop a theory of Finite Element Systems, for the purpose of discretizing sections of vector bundles, in particular those arizing in the theory of elasticity. In the presence of curvature we prove a discrete Bianchi identity. In the flat case we prove a de Rham theorem on cohomology groups. We check that some known mixed finite elements for the stress-displacement formulation of elasticity fit our framework. We also define, in dimension two, the first conforming finite element spaces of metrics with good linearized curvature, corresponding to strain tensors with Saint-Venant compatibility conditions. Cochains with coefficients in rigid motions are given a key role in relating continuous and discrete elasticity complexes.