<table><caption>Table 10: $L^2$ norm of the initial divergence when the initial condition is computed as in Remark 6 for the different meshes of Figure 3 and for degree 0, 1 and 2.</caption><thead><tr><th>degree</th><th>0</th><th>1</th><th>2</th></tr></thead><tbody><tr><td>Triangular</td><td>0</td><td>2.9602e-14</td><td>9.43413e-14</td></tr><tr><td>Cartesian</td><td>0</td><td>0</td><td>0</td></tr><tr><td>Unstructured quad</td><td>0</td><td>4.26275e-14</td><td>1.69086e-13</td></tr></tbody></table>

<table><caption>Table 11: Errors obtained for the test case of the Rotating regular magnetic loop on triangular and quadrangular meshes. The error obtained is close of the optimal rate of convergence.</caption><thead><tr><th rowspan="3">degree</th><th colspan="6">Cartesian</th><th colspan="6">Triangle</th></tr><tr><th colspan="3">u<sub>x</sub></th><th colspan="3">u<sub>y</sub></th><th colspan="3">u<sub>x</sub></th><th colspan="3">u<sub>y</sub></th></tr><tr><th>h</th><th>Error</th><th>rate</th><th>Error</th><th>rate</th><th>h</th><th>Error</th><th>rate</th><th>Error</th><th>rate</th><th>h</th><th>Error</th><th>rate</th></tr></thead><tbody><tr><td rowspan="4">degree 0</td><td>1.00e-01</td><td>6.15e-03</td><td></td><td>5.84e-03</td><td></td><td>8.44e-02</td><td>6.71e-03</td><td></td><td>5.43e-03</td><td></td><td>1.00e-01</td><td>6.15e-03</td><td></td></tr><tr><td>5.00e-02</td><td>5.07e-03</td><td>0.28</td><td>4.57e-03</td><td>0.35</td><td>4.19e-02</td><td>4.72e-03</td><td>0.50</td><td>4.67e-03</td><td>0.22</td><td>5.00e-02</td><td>5.07e-03</td><td>0.28</td></tr><tr><td>2.50e-02</td><td>3.15e-03</td><td>0.69</td><td>2.83e-03</td><td>0.69</td><td>2.23e-02</td><td>2.84e-03</td><td>0.81</td><td>2.72e-03</td><td>0.85</td><td>2.50e-02</td><td>3.15e-03</td><td>0.69</td></tr><tr><td>1.25e-02</td><td>1.69e-03</td><td>0.90</td><td>1.50e-03</td><td>0.92</td><td>1.09e-02</td><td>1.61e-03</td><td>0.80</td><td>1.49e-03</td><td>0.84</td><td>1.25e-02</td><td>1.69e-03</td><td>0.90</td></tr><tr><td rowspan="4">degree 1</td><td colspan="6">Cartesian</td><td colspan="6">Triangle</td></tr><tr><th colspan="3">u<sub>x</sub></th><th colspan="3">u<sub>y</sub></th><th colspan="3">u<sub>x</sub></th><th colspan="3">u<sub>y</sub></th></tr><tr><th>h</th><th>Error</th><th>rate</th><th>Error</th><th>rate</th><th>h</th><th>Error</th><th>rate</th><th>Error</th><th>rate</th><th>h</th><th>Error</th><th>rate</th></tr><tr><td>1.00e-01</td><td>2.42e-03</td><td></td><td>2.47e-03</td><td></td><td>8.44e-02</td><td>3.80e-03</td><td></td><td>3.85e-03</td><td></td><td>1.00e-01</td><td>2.42e-03</td><td></td></tr><tr><td>5.00e-02</td><td>1.29e-03</td><td>0.90</td><td>8.24e-04</td><td>1.58</td><td>4.19e-02</td><td>1.94e-03</td><td>0.96</td><td>1.48e-03</td><td>1.37</td><td>5.00e-02</td><td>1.29e-03</td><td>0.90</td></tr><tr><td>2.50e-02</td><td>6.01e-04</td><td>1.11</td><td>1.93e-04</td><td>2.10</td><td>2.23e-02</td><td>1.02e-03</td><td>1.01</td><td>4.96e-04</td><td>1.73</td><td>2.50e-02</td><td>6.01e-04</td><td>1.11</td></tr><tr><td>1.25e-02</td><td>2.26e-04</td><td>1.41</td><td>4.17e-05</td><td>2.21</td><td>1.09e-02</td><td>3.11e-04</td><td>1.67</td><td>1.71e-04</td><td>1.49</td><td>1.25e-02</td><td>2.26e-04</td><td>1.41</td></tr><tr><td rowspan="4">degree 2</td><td colspan="6">Cartesian</td><td colspan="6">Triangle</td></tr><tr><th colspan="3">u<sub>x</sub></th><th colspan="3">u<sub>y</sub></th><th colspan="3">u<sub>x</sub></th><th colspan="3">u<sub>y</sub></th></tr><tr><th>h</th><th>Error</th><th>rate</th><th>Error</th><th>rate</th><th>h</th><th>Error</th><th>rate</th><th>Error</th><th>rate</th><th>h</th><th>Error</th><th>rate</th></tr><tr><td>1.00e-01</td><td>1.57e-03</td><td></td><td>1.20e-03</td><td></td><td>8.44e-02</td><td>3.21e-03</td><td></td><td>2.29e-03</td><td></td><td>1.00e-01</td><td>1.57e-03</td><td></td></tr><tr><td>5.00e-02</td><td>3.39e-04</td><td>2.21</td><td>2.09e-04</td><td>2.52</td><td>4.19e-02</td><td>8.45e-04</td><td>1.91</td><td>5.59e-04</td><td>2.01</td><td>5.00e-02</td><td>3.39e-04</td><td>2.21</td></tr><tr><td>2.50e-02</td><td>5.53e-05</td><td>2.62</td><td>2.59e-05</td><td>3.01</td><td>2.23e-02</td><td>1.32e-04</td><td>2.94</td><td>8.35e-05</td><td>3.01</td><td>2.50e-02</td><td>5.53e-05</td><td>2.62</td></tr><tr><td>1.25e-02</td><td>5.29e-06</td><td>3.39</td><td>2.41e-06</td><td>3.42</td><td>1.09e-02</td><td>2.43e-05</td><td>2.37</td><td>1.56e-05</td><td>2.35</td><td>1.25e-02</td><td>5.29e-06</td><td>3.39</td></tr></tbody></table>

The computations are led with the approximation space $\mathcal{D}^{\text{curl}}$, and with the Godunov' flux. The initial condition is computed with the method explained in Remark 6.

### 6.3.1 Conservation of the divergence free field

We first check that the divergence of $u$ is preserved, equal to 0. For this, the test case is run until time $t = \pi$ on the different meshes shown in Figure 3. The initial $L^2$ norm of $\nabla^*u^0$ is summarized in Table 10. Results show that the method described in Remark 6 ensures the zero divergence for the initial condition.

The difference with respect to the initial divergence is plotted in Figure 10, and shows preservation up to round-off errors of the divergence free field.

### 6.3.2 Convergence test

We wish now to perform a convergence test on the series of Cartesian and unstructured triangular meshes described in the beginning of the section. In this test, the computation is led until $t = 0.5$. The convergence curves for the two variables $u_x$ and $u_y$ are plotted in Figure 11, whereas the convergence rate computations are shown in Table 11. Both the table and the figure represent a convergence close of the optimal order of convergence.
<figure><img src="image_13.png" /></figure>

Divergence free initialization,
Lax-Friedrich flux.

<figure><img src="image_14.png" /></figure>

No divergence free initialization,
flux (22).

<figure><img src="image_15.png" /></figure>

Divergence free initialization,
flux (22), $t_{final} = 2$.

<figure><img src="image_16.png" /></figure>

Divergence free initialization,
flux (22), $t_{final} = 20$.

Figure 10: Adimensioned magnetic energy obtained for the test case *Discontinuous magnetic field loop*. On the top left picture, results obtained with a divergence free initialization and the Lax-Friedrich flux show a stable behaviour. On the top right picture, results obtained with a non divergence free initialization and the flux (22) show an unstable behaviour except for the degree 0 scheme. On the bottom picture, results obtained with a divergence free initialization and the flux (22) show a stable behaviour, for both final time 2 on the left and final time 20 on the right.
<figure><img src="image_15.png" /><figcaption>Figure 10: Time evolution of $||∇^*u − ∇^*u^0||_2$ with respect to the time for a coarse Cartesian, unstructured quadrangular and a triangular mesh (all shown in Figure 3). Note that the scale is $10^{-11}$, so that the figure shows exact (up to round-off errors) preservation of the divergence.</figcaption></figure>
solution, but it does not include any diffusion, and so no mechanism for damping the round-off errors around 0. For very long time integration we foresee that the accumulation of round-off errors may lead to a non divergence free vector, which may itself lead to instability of the numerical scheme, similar to the one observed on top right of the Figure 10. This potential instability could be cured by performing a divergence cleaning at some time steps, similar to redistanciation in level-set methods, or to re-orthogonalization in Krylov methods.

# 6 Conclusion

In this article, the discrete preservation of differential constraints was investigated for the discontinuous Galerkin methods. The method developed within this article relies on a previously proposed framework for deriving appropriate approximation spaces for vectors, which fits in a discrete de-Rham context [71]. For a continuous equation on a vector that preserves either the curl or the divergence, we were able to prove that the classical discontinuous Galerkin method preserves also this constraint at the discrete level under the following assumptions:

* Use the correct approximation space for vector unknowns (for straight triangular meshes, the approximation space is the classical one)

* Use a numerical flux which diffusion is either parallel (for preserving a curl) or orthogonal (for preserving a divergence) to the normal of the faces.

Notably, the global curl or the global divergence that are preserved are defined in an adjoint sense. This use of the adjoint de-Rham complex is in fact very well suited with the fact that our numerical method is of Galerkin's type. Indeed, taking a strong derivative (divergence or curl) of the solution of a numerical scheme seems to be a tedious task in general. On the contrary, taking the adjoint of the exterior derivative is a very simple operation for a Galerkin based numerical scheme: it just consists in testing the weak formulation with a gradient or a curl of a function. For the induction equation, it was even possible to derive a discrete transport equation in the space $A_{k+1}$ for the adjoint divergence, starting from the discontinuous Galerkin method in $dB_k^{\text{curl}}$ for the unknown vector. Last, the clear discrete de-Rham framework that was developed in this article allows to initialize easily a divergence free or a curl free field from the knowledge of its potential.

Numerical tests were performed on three linear systems: the two dimensional Maxwell system for the conservation of the divergence, the two dimensional wave system for the conservation of the curl, and the two dimensional induction equation. Numerical tests confirmed the theoretical results, and show also that the theoretical results seem to be sharp, in the sense that if one of the hypothesis of Proposition 3 and Proposition 2 is not ensured (wrong approximation space or diffusion not in the right direction), the conservation of the curl or the divergence is jeopardized. Last, the implementation was done with a special family of finite elements on quads (and also on triangles), but theoretical results hold for other finite elements proposed in [71].

Note that the numerical tests were performed only on linear systems, but Proposition 2 and Proposition 3 hold also for nonlinear problems. Still, these propositions ensure only geometrical properties on the solution, but do not say anything on the stability of the schemes. As our scheme is fully collocated, we hope to avoid the difficulty of multidimensional Riemann problems [33, 22, 7] raised by the staggering of the magnetic field. Still, the extension to nonlinear systems will lead to the adaptation of the necessary slope or flux limiting to our framework, which is an opened question for the moment. Concerning divergence free problems, efficiency of the method could be improved by considering the same numerical method, but with a trimmed basis consisting of the cell divergence free basis as in [23]. Note however that for conforming with the rest of the method, the basis should then be divergence free in the adjoint sense. Another aspect of the nonlinear problems is the bound preserving property for the MHD system, which was recently resolved in [85], and which is strongly linked with the divergence free condition [84]. It would be interesting to check whether our method, which is divergence free preserving, but in the adjoint sense, fits in the theory developed in [84, 85] for the bound preservation. Concerning curl free methods,
<figure><img src="image_16.png" /><figcaption>Figure 11: Error curves for the test case described in subsection 6.3.2. The top figures are the error on $u_x$, whereas the bottom figures are the error on $u_y$. The left column matches with triangular meshes whereas the right column is for Cartesian meshes.</figcaption></figure>

# 7 Conclusion

In this article, the discrete preservation of differential constraints was investigated for the discontinuous Galerkin methods. The method developed within this article relies on a previously proposed framework for deriving appropriate approximation spaces for vectors, which fits in a discrete de-Rham context [54]. Based on this framework, we were able to propose a discrete version of the Hodge star operators, exterior derivative and the adjoint of the exterior derivative. For a continuous equation on a vector that preserves either the curl or the divergence, we were able to prove that the classical discontinuous Galerkin method preserves also this constraint at the discrete level under the following assumptions:

* Use the correct approximation space for vector unknowns

* Use a numerical flux which diffusion is either parallel (for preserving a curl) or orthogonal (for preserving a divergence) to the normal of the faces.

Notably, the curl or the divergence that are preserved are defined in an adjoint sense, namely they are exterior co-derivative. This use of the adjoint de-Rham complex is in fact very well suited with the fact that our numerical method is of Galerkin' type. Indeed, taking the strong exterior derivative of the solution of a numerical scheme seems to be a tedious task in general. On the contrary, taking the adjoint of the exterior derivative is a very simple operation for a Galerkin based numerical scheme: it just consists in testing the weak formulation with a gradient or a curl of a function. For the induction equation, it was even possible to derive a discrete transport equation in the space $A_{k+1}$ for the adjoint divergence, starting from the discontinuous Galerkin method in $dB_k^{\text{curl}}$ for the unknown vector. Last, the clear discrete de-Rham framework that was developed in this article allows to initialize easily a divergence free or a curl free field from the knowledge of its potential.
applications to the hyperbolic version of the Hamilton-Jacobi method [55] could be considered in the future, and as this is a problem with a curl free field, a trimmed adjoint curl free basis function could be considered as in [57] (but in our case, still in the adjoint sense).

Last, it is also important to see that Proposition 2 and Proposition 3 hold also in dimension 3, provided the right approximation spaces were derived. We are currently investigating the extension of the results of this article to dimension 3 and to nonlinear systems of conservation law.

## References

[1] Douglas Norman Arnold. *Finite element exterior calculus*. SIAM, 2018.

[2] Douglas Norman Arnold, Richard Steven Falk, and Ragnar Winther. Finite element exterior calculus, homological techniques, and applications. *Acta numerica*, 15:1–155, 2006.

[3] Douglas Norman Arnold, Richard Steven Falk, and Ragnar Winther. Finite element exterior calculus: from Hodge theory to numerical stability. *Bulletin of the American mathematical society*, 47(2):281–354, 2010.

[4] Douglas Norman Arnold and Anders Bernhard Logg. Periodic table of the finite elements. *SIAM News*, 47(9):212, 2014.

[5] Dinshaw S. Balsara. Divergence-free adaptive mesh refinement for magnetohydrodynamics. *Journal of Computational Physics*, 174(2):614–648, 2001.

[6] Dinshaw S. Balsara. Second-order-accurate schemes for magnetohydrodynamics with divergence-free reconstruction. *The Astrophysical Journal Supplement Series*, 151(1):149, 2004.

[7] Dinshaw S. Balsara, Roger Käppeli, Walter Boscheri, and Michael Dumbser. Curl constraint-preserving reconstruction and the guidance it gives for mimetic scheme design. *Communications on Applied Mathematics and Computation*, pages 1–60, 2021.

[8] Dinshaw S. Balsara and Daniel Shields Spicer. A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations. *Journal of Computational Physics*, 149(2):270–292, 1999.

[9] Wasilij Barsukow. Stationarity preserving schemes for multi-dimensional linear systems. *Mathematics of Computation*, 88(318):1621–1645, 2019.

[10] Wasilij Barsukow. Truly multi-dimensional all-speed schemes for the Euler equations on Cartesian grids. *Journal of Computational Physics*, 435:110216, 2021.

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Numerical tests were performed on three linear systems: the two dimensional Maxwell system for the conservation of the divergence, the two dimensional wave system for the conservation of the curl, and the two dimensional induction equation. Numerical tests confirmed the theoretical results, and show also that the theoretical results seem to be sharp, in the sense that if one of the hypothesis of Proposition 5 and Proposition 4 is not ensured (wrong approximation space or diffusion not in the right direction), the conservation of the curl or the divergence is jeopardized.

Note that the numerical tests were performed only on linear systems, but the propositions that were proven hold also for nonlinear problems. Also, the implementation was done with a special family of finite elements on quads (and also on triangles), but theoretical results hold for other finite elements proposed in [54].

Last, it is also important to see that Proposition 4 and Proposition 5 hold also in dimension 3, provided the right approximation spaces were derived. We are currently investigating the extension of the results of this article to dimension 3 and to nonlinear systems of conservation law.

## References

[1] Douglas Norman Arnold. *Finite element exterior calculus*. SIAM, 2018.

[2] Douglas Norman Arnold, Richard Steven Falk, and Ragnar Winther. Finite element exterior calculus, homological techniques, and applications. *Acta numerica*, 15:1–155, 2006.

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# A Summary of the operators on differential forms and on their proxies

In the following table, the proxy of the different operators used in this article are summarized, when the choice of proxy for $\Lambda^1$ is (1).

<table><thead><tr><th>k</th><th>0</th><th>1</th><th>2</th></tr></thead><tbody><tr><td>Element of Λ<sup>k</sup></td><td>f</td><td>u</td><td>f</td></tr><tr><td>Proxy</td><td>f</td><td>u</td><td>f</td></tr><tr><td>Proxy(★<sub>{k}</sub>)</td><td>f</td><td>u<sup>⊥</sup></td><td>f</td></tr><tr><td>Proxy(d<sup>k</sup>)</td><td>∇f</td><td>∇<sup>⊥</sup>·u</td><td>—</td></tr><tr><td>Proxy(δ<sup>k</sup>)</td><td>—</td><td>—∇·u</td><td>—∇<sup>⊥</sup>f</td></tr><tr><td>Proxy(δ<sup>k+1</sup>d<sup>k</sup> + d<sup>k-1</sup>δ<sup>k</sup>)</td><td>—∇·(∇f)</td><td>—∇(∇·u) — ∇<sup>⊥</sup>(∇<sup>⊥</sup>·u)</td><td>—∇<sup>⊥</sup>·(∇<sup>⊥</sup>f)</td></tr><tr><td>Proxy(i<sub>b</sub>)</td><td>—</td><td>b·u</td><td>b<sup>⊥</sup>f</td></tr><tr><td>Proxy(j<sub>b</sub>)</td><td>fb</td><td>—u<sup>⊥</sup>·b</td><td>—</td></tr><tr><td>Proxy(L<sub>b</sub>)</td><td>b·∇f</td><td>∇(b·u) + b<sup>⊥</sup>∇<sup>⊥</sup>·u</td><td>∇·(fb)</td></tr><tr><td>Proxy(L<sub>b</sub>)</td><td>—∇·(fb)</td><td>—∇<sup>⊥</sup>(det(b, u)) — b∇·u</td><td>—b·∇f</td></tr></tbody></table>

In the next table, the proxy of the different operators used in this article are summarized, when the choice of proxy for $\Lambda^1$ is (2).

<table><thead><tr><th>k</th><th>0</th><th>1</th><th>2</th></tr></thead><tbody><tr><td>Element of Λ<sup>k</sup></td><td>f</td><td>u</td><td>f</td></tr><tr><td>Proxy<sup>⊥</sup></td><td>f</td><td>u</td><td>f</td></tr><tr><td>Proxy<sup>⊥</sup>(★<sub>{k}</sub>)</td><td>f</td><td>u<sup>⊥</sup></td><td>f</td></tr><tr><td>Proxy<sup>⊥</sup>(d<sup>k</sup>)</td><td>∇<sup>⊥</sup>f</td><td>—∇·u</td><td>—</td></tr><tr><td>Proxy<sup>⊥</sup>(δ<sup>k</sup>)</td><td>—</td><td>—∇<sup>⊥</sup>·u</td><td>∇f</td></tr><tr><td>Proxy<sup>⊥</sup>(δ<sup>k+1</sup>d<sup>k</sup> + d<sup>k-1</sup>δ<sup>k</sup>)</td><td>—∇<sup>⊥</sup>·(∇<sup>⊥</sup>f)</td><td>—∇(∇·u) — ∇<sup>⊥</sup>(∇<sup>⊥</sup>·u)</td><td>—∇·(∇f)</td></tr><tr><td>Proxy<sup>⊥</sup>(i<sub>b</sub>)</td><td>—</td><td>det(b, u)</td><td>—fb</td></tr><tr><td>Proxy<sup>⊥</sup>(j<sub>b</sub>)</td><td>fb<sup>⊥</sup></td><td>—u·b</td><td>—</td></tr><tr><td>Proxy<sup>⊥</sup>(L<sub>b</sub>)</td><td>b·∇f</td><td>∇<sup>⊥</sup>(det(b, u)) + b∇·u</td><td>∇·(fb)</td></tr><tr><td>Proxy<sup>⊥</sup>(L<sub>b</sub>)</td><td>—∇·(fb)</td><td>—∇(b·u) — b<sup>⊥</sup>∇<sup>⊥</sup>·u</td><td>—b·∇f</td></tr></tbody></table>

Last, the effect of the exterior product on the proxies does not depend on the choice of proxies, and was summarized in Table 1.