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A polyhedral discrete de Rham numerical scheme for the Yang-Mills equations

Jérôme Droniou, Todd A. Oliynyk, Jia Jia Qian

TL;DR

The paper addresses robust numerical discretisation of the nonlinear Yang–Mills equations on generic polyhedral meshes. It introduces a fully discrete, Lie algebra–valued DDR (LADDR) framework with a constrained weak formulation that uses a Lagrange multiplier to enforce the nonlinear constraint exactly at the discrete level, alongside carefully designed discretisations of the Lie brackets to preserve Ad-invariance. Key contributions include energy estimates for the scheme, exact constraint preservation in the constrained formulation, demonstration of convergence and constraint preservation in 3D tests, and a clear pathway to higher-order extensions. The approach offers flexibility with arbitrary mesh types and orders, and shows promise for applications beyond Yang–Mills, including potential extensions to Einstein-type equations and VEM-compatible polytopal methods.

Abstract

We present a discretisation of the 3+1 formulation of the Yang-Mills equations in the temporal gauge, using a Lie algebra-valued extension of the discrete de Rham (DDR) sequence, that preserves the non-linear constraint exactly. In contrast to Maxwell's equations, where the preservation of the analogous constraint only depends on reproducing some complex properties of the continuous de Rham sequence, the preservation of the non-linear constraint relies for the Yang-Mills equations on a constrained formulation, previously proposed in [10]. The fully discrete nature of the DDR method requires to devise appropriate constructions of the non-linear terms, adapted to the discrete spaces and to the need for replicating the crucial Ad-invariance property of the $L^2$-product. We then prove some energy estimates, and provide results of 3D numerical simulations based on this scheme.

A polyhedral discrete de Rham numerical scheme for the Yang-Mills equations

TL;DR

The paper addresses robust numerical discretisation of the nonlinear Yang–Mills equations on generic polyhedral meshes. It introduces a fully discrete, Lie algebra–valued DDR (LADDR) framework with a constrained weak formulation that uses a Lagrange multiplier to enforce the nonlinear constraint exactly at the discrete level, alongside carefully designed discretisations of the Lie brackets to preserve Ad-invariance. Key contributions include energy estimates for the scheme, exact constraint preservation in the constrained formulation, demonstration of convergence and constraint preservation in 3D tests, and a clear pathway to higher-order extensions. The approach offers flexibility with arbitrary mesh types and orders, and shows promise for applications beyond Yang–Mills, including potential extensions to Einstein-type equations and VEM-compatible polytopal methods.

Abstract

We present a discretisation of the 3+1 formulation of the Yang-Mills equations in the temporal gauge, using a Lie algebra-valued extension of the discrete de Rham (DDR) sequence, that preserves the non-linear constraint exactly. In contrast to Maxwell's equations, where the preservation of the analogous constraint only depends on reproducing some complex properties of the continuous de Rham sequence, the preservation of the non-linear constraint relies for the Yang-Mills equations on a constrained formulation, previously proposed in [10]. The fully discrete nature of the DDR method requires to devise appropriate constructions of the non-linear terms, adapted to the discrete spaces and to the need for replicating the crucial Ad-invariance property of the -product. We then prove some energy estimates, and provide results of 3D numerical simulations based on this scheme.
Paper Structure (24 sections, 5 theorems, 72 equations, 9 figures, 1 table)

This paper contains 24 sections, 5 theorems, 72 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Yang--Mills equations in flat $3+1$ formalism
  3. Lowest order Discrete de Rham complex
  4. Notations
  5. Scalar DDR complex
  6. Spaces and discrete vector calculus operators
  7. Potential reconstructions and inner products
  8. Lie algebra-valued DDR complex
  9. Spaces and extended operators
  10. Schemes
  11. Maxwell equations
  12. Yang--Mills equations
  13. Weak formulation
  14. Constrained weak formulation
  15. Discretisation of the Lie brackets
  16. ...and 9 more sections

Key Result

Proposition 2

If $(\boldsymbol{A},\boldsymbol{E})$ solve eq:ym.weak then for all $q\in H^1(U)\otimes\mathfrak{g}$ the quantity is constant in time. In particular, if the constraint eq:constraint.ym is satisfied at $t=0$, then it is satisfied for all $t\in [0,T]$.

Figures (9)

  • Figure 1: "Voro-small-0" mesh
  • Figure 2: "Tetgen-Cube-0" mesh
  • Figure 3: "Cubic-Cells" mesh
  • Figure 5: "Voro-small-0" mesh
  • Figure 6: "Tetgen-Cube-0" mesh
  • ...and 4 more figures

Theorems & Definitions (13)

  • Remark 1: Simpler presentation of the spaces
  • Proposition 2: Preservation of constraint for the weak Yang--Mills equations
  • proof
  • Remark 3: Alternative approaches
  • Lemma 4: Relationship between weak and constrained weak formulation
  • proof
  • Remark 5: About the Lagrange multiplier
  • Proposition 6: Energy conservation
  • proof
  • Proposition 7: Constraint preservation
  • ...and 3 more