# Discontinuous Galerkin methods for hyperbolic system that preserve exactly a curl or a divergence constraint

Jonathan Jung, Vincent Perrier

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Jonathan Jung, Vincent Perrier. Discontinuous Galerkin methods for hyperbolic system that preserve exactly a curl or a divergence constraint. ECCOMAS 2024 - The 9th European Congress on Computational Methods in Applied Sciences and Engineering, Jun 2024, Lisbonne, Portugal. hal-04884111

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Submitted on 13 Jan 2025

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# Discontinuous Galerkin methods for hyperbolic system that preserve exactly a curl or a divergence constraint

Jonathan Jung¹, Vincent Perrier²

¹ LMA, E2S-UPPA, and Cagire team, Inria Bordeaux Sud-Ouest, Avenue de l'Université, 64013 Pau Cedex, France. jonathan.jung@univ-pau.fr.

² Cagire team, Inria Bordeaux Sud-Ouest and LMA, E2S-UPPA, Avenue de l'Université, 64013 Pau Cedex, France. vincent.perrier@inria.fr.

**Keywords:** Finite element method, discontinuous Galerkin method, discrete de-Rham complex, structure preservation.

This talk will be dedicated to the discontinuous Galerkin approximation method for hyperbolic systems. It will be focused on the preservation of differential constraint that is not part of the system; for example, if the wave system is considered

$$ \left\{ \begin{array}{l} \partial_t p + \operatorname{div}_x u = 0 \\ \partial_t u + \nabla_x p = 0 \end{array} \right. , \qquad (1) $$

then taking the curl of the second equation gives

$$ \partial_t (\operatorname{curl}_x u) = 0. \qquad (2) $$

Another example is the following two-dimensional form of the Maxwell system

$$ \left\{ \begin{array}{l} \partial_t e + \nabla_x^\perp b = 0 \\ \partial_t b + \nabla_x^\perp \cdot e = 0 \end{array} \right. , \qquad (3) $$

for which the following implicit constraint holds

$$ \partial_t (\nabla_x^\perp \cdot e) = 0. \qquad (4) $$

Usually, the classical discontinuous Galerkin method designed for (1) will not preserve the additional differential constraint (2), and a discontinuous Galerkin method designed for (3) will not preserve (4). Usual cure to this rely on the use of the conforming discrete de-Rham finite element space complex designed in [8, 9], see also [4, 1] for a more modern presentation, which can be summarized by the two following two-dimensional diagrams

$$ \left\{ \begin{array}{rcl} \mathbb{P}_{k+1} & \xrightarrow{\nabla_x^\perp} & \mathbb{R}\mathbb{T}_{k+1} \\ \mathbb{P}_{k+1} & \xrightarrow{\nabla_x} & \mathbb{N}_{k+1} \end{array} \right. \xrightarrow{\nabla_x^\perp} \mathbb{d}\mathbb{P}_k \qquad (5) $$

where $\mathbb{P}_{k+1}$, $\mathbb{R}\mathbb{T}_{k+1}$, $\mathbb{N}_{k+1}$ and $\mathbb{d}\mathbb{P}_k$ are respectively the classical continuous finite element space, the Raviart-Thomas finite element space, the two-dimensional Nédélec finite element space and the discontinuous finite element space. Using these diagrams or their
variants was proposed for example in [2, 3, 7, 10] for ensuring the differential constraints,
but this results in a numerical method where the degrees of freedom are spread on all the
entities of the mesh (not only cells, but also faces and edges). This leads also to concerns
about the exact form of the numerical diffusion, which is located on faces for the classical
discontinuous Galerkin method.

In this talk, we will propose to replace the vector spaces of (5) by discontinuous finite
element spaces designed for ensuring an alternative discrete de-Rham complex, on both
triangular and quadrangular meshes. From this, we will be able to derive discontinuous
Galerkin methods that ensure the preservation of the differential constraints. This work
can be seen as the extension to quads and to higher order of the properties previously
observed in [5, 6].

REFERENCES

[1] DOUGLAS NORMAN ARNOLD, *Finite elements exterior calculus*. SIAM, 2018.

[2] DINSHAW S. BALSARA AND DANIEL S. SPICER, *A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamical simulations*, Journal of Computational Physics, 149 (2), 270–292, 1999.

[3] W. BOSCHERI, M. DUMBSER, M. IORIATTI, I. PESHKOV, AND E. ROMENSKI, *A structure-preserving staggered semi-implicit finite volume scheme for continuum mechanics*, Journal of Computational Physics, 424 (2021).

[4] ALEXANDRE ERN AND JEAN-LUC GUERMOND, *Finite elements I: Approximation and interpolation*. Springer, 2021.

[5] HERVÉ GUILLARD, *On the behavior of upwind schemes in the low Mach number limit*. IV: $P_0$ approximation on triangular and tetrahedral cells*, Computers & Fluids, 38 (2009), pp. 1969–1972.

[6] JONATHAN JUNG AND VINCENT PERRIER, *Behaviour of the discontinuous Galerkin method for compressible flows at low Mach number on triangles and tetrahedrons*, SIAM Journal on Scientific Computing, 2023, accepted.

[7] PATRICK MULLEN, ALEXANDER MCKENZIE, DMITRY PAVLOV, LUKE DURANT, YIYING TONG, EVA KANSO, JERROLD E. MARSDEN, AND MATHIEU DESBRUN, *Discrete Lie advection of differential forms*, Foundations of Computational Mathematics, 11, pp.131–149, 2011.

[8] JEAN-CLAUDE NÉDÉLEC, *Mixed finite elements in $\mathbb{R}^3*$. Numerische Mathematik, 35, 315–341, 1980.

[9] PIERRE-ARNAUD RAVIART AND JEAN-MARIE THOMAS, *A mixed finite element method for 2-nd order elliptic problems*. in Mathematical Aspects of Finite Element Methods: Proceedings of the Conference Held in Rome, December 10–12, 1975.

[10] MAURIZIO TAVELLI AND MICHAEL DUMBSER, *A pressure-based semi-implicit space-time discontinuous Galerkin method on staggered unstructured meshes for the solution of the compressible Navier-Stokes equations at all Mach numbers*, Journal of Computational Physics, 341 (2017), pp. 341–376.