Finite element conformal complexes in three dimensions

TL;DR

This paper develops finite element conformal complexes in three dimensions using an extended BGG framework, enabling stable numerical methods for complex physical applications involving conformal tensors.

Contribution

It introduces a novel application of the discrete BGG framework combined with geometric decomposition to construct simpler, local bubble finite element complexes in 3D.

Findings

Constructed finite element conformal Hessian and elasticity complexes in 3D.

Supported stable, structure-preserving numerical methods for relativity and elasticity.

Provided a systematic approach to develop complexes with varying smoothness levels.

Abstract

This paper extends the Bernstein-Gelfand-Gelfand (BGG) framework to the construction of finite element conformal Hessian complexes and conformal elasticity complexes in three dimensions involving conformal tensors (i.e., symmetric and traceless tensors). These complexes incorporate higher-order differential operators, including the linearized Cotton-York operator, and require conformal tensor spaces with nontrivial smoothness and trace conditions. A novel application of the discrete BGG framework, combined with the geometric decomposition of bubble spaces and a reduction operation, to local bubble finite element complexes is introduced. This yields simpler and more tractable constructions than global BGG-based approaches, and leads to the bubble conformal complexes. Building on these bubble conformal complexes and the associated face bubble complexes, finite element conformal Hessian complexes and conformal elasticity complexes with varying degrees of smoothness are systematically developed. The resulting complexes support stable and structure-preserving numerical methods for applications in relativity, Cosserat elasticity, and fluid mechanics.