Conforming Discrete Gradgrad-Complexes in Three Dimensions

TL;DR

This paper constructs the first conforming discrete 3D Gradgrad-complexes with finite element spaces, which are exact and applicable to the linearized Einstein-Bianchi system.

Contribution

It introduces the first family of conforming discrete 3D Gradgrad-complexes with exactness, expanding finite element methods for complex systems.

Findings

Discrete complexes are exact with kernel properties.

Applicable to mixed formulations of Einstein-Bianchi system.

First construction of conforming 3D Gradgrad-complexes.

Abstract

In this paper, the first family of conforming discrete three dimensional Gradgrad-complexes consisting of finite element spaces is constructed. These discrete complexes are exact in the sense that the range of each discrete map is the kernel space of the succeeding one. These spaces can be used in the mixed form of the linearized Einstein-Bianchi system.