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

Marien-Lorenzo Hanot, Kaibo Hu

TL;DR

The paper reformulates the linearized ADM Einstein equations as a Hodge-Dirac wave system on the div-div complex, enabling structure-preserving discretization via finite element exterior calculus. It proves continuous well-posedness, develops a variational framework that accommodates conforming and non-conforming discrete complexes, and provides rigorous error estimates combining spatial and temporal discretization effects. A concrete backward-Euler scheme on a tensor-product spline complex is analyzed and validated through numerical experiments that demonstrate expected convergence behavior and constraint preservation. The approach offers robust gauge handling and constraint enforcement, with potential to extend to nonlinear relativity problems. Overall, it presents a principled, mathematically grounded path to stable, long-time simulations in numerical relativity using FEEC-based discretizations.

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.

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

TL;DR

The paper reformulates the linearized ADM Einstein equations as a Hodge-Dirac wave system on the div-div complex, enabling structure-preserving discretization via finite element exterior calculus. It proves continuous well-posedness, develops a variational framework that accommodates conforming and non-conforming discrete complexes, and provides rigorous error estimates combining spatial and temporal discretization effects. A concrete backward-Euler scheme on a tensor-product spline complex is analyzed and validated through numerical experiments that demonstrate expected convergence behavior and constraint preservation. The approach offers robust gauge handling and constraint enforcement, with potential to extend to nonlinear relativity problems. Overall, it presents a principled, mathematically grounded path to stable, long-time simulations in numerical relativity using FEEC-based discretizations.

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.

Paper Structure

This paper contains 14 sections, 17 theorems, 78 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Reformulation of the ADM formulation
  3. Continuous well-posedness
  4. Variational formulation and discretization
  5. Spatial discretization
  6. Time discretization
  7. Proof of the error estimate
  8. Examples: a specific scheme
  9. Numerical results
  10. Discrete complex
  11. Wave-like solutions
  12. Results
  13. Conclusions
  14. Vector calculus identities

Key Result

Theorem 1

The solution $\gamma$ of eq:HDADM with initial conditions $(A_0,\gamma_0)$ sastifies the linearized York version of the ADM formulation eq:def.York.ADM for the gauge $\alpha := \frac{1}{2} \mathop{\mathrm{tr}}\nolimits \gamma$ and $\beta := \frac{1}{2} \int_0^t \mathop{\mathrm{\bf div}}\nolimits S\g

Figures (3)

  • Figure 1: Error with respect to the mesh size for various time steps using the Euler time stepping.
  • Figure 2: Error with respect to the mesh size for various time steps using the Crank-Nicolson time stepping.
  • Figure 3: Evolution of the error during the simulated time for the first case.

Theorems & Definitions (42)

  • Theorem 1
  • proof
  • Remark 2
  • Remark 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Remark 6: Recovering the shift vector
  • Lemma 7: Skew adjointness
  • ...and 32 more