Hodge-Dirac wave systems and structure-preserving discretizations of the linearized Einstein equations

TL;DR

This paper reformulates the linearized Einstein equations as a Hodge-Dirac wave system, enabling structure-preserving discretizations that improve numerical stability and accuracy in general relativity simulations.

Contribution

It introduces a novel Hodge-Dirac wave formulation of the linearized Einstein equations and develops compatible discretization methods with proven error estimates.

Findings

Well-posedness of the Hodge-Dirac wave system established

Discretization scheme applicable to conforming and non-conforming complexes

Error estimates derived under minimal assumptions

Abstract

We derive a reformulation of the linearized Arnowitt-Deser-Misner (ADM) equations as a Hodge-Dirac wave system with the divdiv complex, addressing challenges in numerical relativity such as gauge fixing, constraint propagation, and tensor symmetries. The differential and algebraic structures of the divdiv complex ensure the well-posedness of the formulation and facilitate structure-preserving discretization via finite element exterior calculus. We establish the well-posedness of this Hodge-Dirac wave equation and develop a discretization scheme applicable to both conforming and non-conforming discrete complexes, deriving error estimates under minimal assumptions.