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

Droniou, Jérôme; Oliynyk, Todd A.; Qian, Jia Jia

doi:10.1016/j.jcp.2023.111955

Cited by 4 publications

(2 citation statements)

References 20 publications

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“…Despite their non-conformity, polytopal technologies can be used to develop compatible frameworks. Polytopal discretisations of the de Rham complex (1.1) have been proposed, e.g., in [10,33,38], and applied to a variety of models , such as magnetostatics [8,34], the Stokes equations [11], and the Yang-Mills equations [47]; they have also inspired further developments, based on the same principles, for other complexes of interest such as variants of the de Rham complex with increased regularity [32,55], elasticity complexes [19,44], and the Stokes complex [12,14,49]. Polytopal complexes have additionally been used to construct methods that are robust with respect to the variations of physical parameters, in particular for the Stokes problem [11], for the Reissner-Mindlin equation [43], or the Brinkman model [33].…”

Section: Introductionmentioning

confidence: 99%

An exterior calculus framework for polytopal methods

Bonaldi¹,

Pietro²,

Droniou³

et al. 2023

Preprint

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We develop in this work the first polytopal complexes of differential forms. These complexes, inspired by the Discrete De Rham and the Virtual Element approaches, are discrete versions of the de Rham complex of differential forms built on meshes made of general polytopal elements. Both constructions benefit from the high-level approach of polytopal methods, which leads, on certain meshes, to leaner constructions than the finite element method. We establish commutation properties between the interpolators and the discrete and continuous exterior derivatives, prove key polynomial consistency results for the complexes, and show that their cohomologies are isomorphic to the cohomology of the continuous de Rham complex.

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Section: Introductionmentioning

confidence: 99%

An exterior calculus framework for polytopal methods

Bonaldi¹,

Pietro²,

Droniou³

et al. 2023

Preprint

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“…Constraint-preserving numerical methods for the Yang-Mills equations, based on the finite element exterior calculus [1], have been developed and investigated in [7,8,4]. In [10], within framework of a discrete de Rham method, another structure-preserving numerical scheme to approximate the Yang-Mills equations has been constructed.…”

Section: Introductionmentioning

confidence: 99%

2D Discrete Hodge–Dirac Operator on the Torus

Sushch

2022

Symmetry

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We discuss a discretization of the de Rham–Hodge theory in the two-dimensional case based on a discrete exterior calculus framework. We present discrete analogues of the Hodge–Dirac and Laplace operators in which key geometric aspects of the continuum counterpart are captured. We provide and prove a discrete version of the Hodge decomposition theorem. The goal of this work is to develop a satisfactory discrete model of the de Rham–Hodge theory on manifolds that are homeomorphic to the torus. Special attention has been paid to discrete models on a combinatorial torus. In this particular case, we also define and calculate the cohomology groups.

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2D Discrete Yang–Mills Equations on the Torus

Sushch

2024

Symmetry

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In this paper, we introduce a discretization scheme for the Yang–Mills equations in the two-dimensional case using a framework based on discrete exterior calculus. Within this framework, we define discrete versions of the exterior covariant derivative operator and its adjoint, which capture essential geometric features similar to their continuous counterparts. Our focus is on discrete models defined on a combinatorial torus, where the discrete Yang–Mills equations are presented in the form of both a system of difference equations and a matrix form.

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

Cited by 4 publications

References 20 publications

An exterior calculus framework for polytopal methods

An exterior calculus framework for polytopal methods

2D Discrete Hodge–Dirac Operator on the Torus

2D Discrete Yang–Mills Equations on the Torus

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