Conforming finite element DIVDIV complexes and the application for the linearized Einstein-Bianchi system

TL;DR

This paper introduces the first conforming finite element divdiv complexes on tetrahedral grids in three dimensions, enabling accurate discretization of the linearized Einstein-Bianchi system for advanced geometric analysis.

Contribution

It constructs new finite element spaces for $H( ext{symcurl})$ and $H^1$ that form an exact complex, extending finite element methods for complex geometric PDEs.

Findings

Finite element spaces for $H( ext{divdiv})$ are based on recent preprint work.

New finite element spaces for $H( ext{symcurl})$ and $H^1$ are constructed and proven to form an exact complex.

The complexes are applied to discretize the linearized Einstein-Bianchi system effectively.

Abstract

This paper presents the first family of conforming finite element divdiv complexes on tetrahedral grids in three dimensions. In these complexes, finite element spaces of $H(\text{divdiv},\Omega;\mathbb{S})$ are from a current preprint [Chen and Huang, arXiv: 2007.12399, 2020] while finite element spaces of both $H(\text{symcurl},\Omega;\mathbb{T})$ and $H^1(\Omega;\mathbb{R}^3)$ are newly constructed here. It is proved that these finite element complexes are exact. As a result, they can be used to discretize the linearized Einstein-Bianchi system within the dual formulation.