Charge-conserving hybrid methods for the Yang–Mills equations
The SMAI journal of computational mathematics, Volume 7 (2021), pp. 97-119.

The Yang–Mills equations generalize Maxwell’s equations to nonabelian gauge groups, and a quantity analogous to charge is locally conserved by the nonlinear time evolution. Christiansen and Winther [8] observed that, in the nonabelian case, the Galerkin method with Lie algebra-valued finite element differential forms appears to conserve charge globally but not locally, not even in a weak sense. We introduce a new hybridization of this method, give an alternative expression for the numerical charge in terms of the hybrid variables, and show that a local, per-element charge conservation law automatically holds.

Published online:
DOI: 10.5802/smai-jcm.73
Classification: 65M60
Keywords: finite element method, domain decomposition, conservation laws, charge conservation, Yang–Mills equations, Maxwell’s equations
Yakov Berchenko-Kogan 1; Ari Stern 2

1 University of Hawaii
2 Washington University in St. Louis
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Yakov Berchenko-Kogan; Ari Stern. Charge-conserving hybrid methods for the Yang–Mills equations. The SMAI journal of computational mathematics, Volume 7 (2021), pp. 97-119. doi : 10.5802/smai-jcm.73. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.73/

[1] M. S. Alnæs; J. Blechta; J. Hake; A. Johansson; B. Kehlet; A. Logg; C. Richardson; J. Ring; M. E. Rognes; G. N. Wells The FEniCS Project Version 1.5, Archive of Numerical Software, Volume 3 (2015) no. 100 | DOI

[2] D. N. Arnold; R. S. Falk; R. Winther Finite element exterior calculus, homological techniques, and applications, Acta Numer., Volume 15 (2006), pp. 1-155 | DOI | MR | Zbl

[3] D. N. Arnold; R. S. Falk; R. Winther Finite element exterior calculus: from Hodge theory to numerical stability, Bull. Am. Math. Soc., Volume 47 (2010) no. 2, pp. 281-354 | arXiv | DOI | MR | Zbl

[4] Y. Berchenko-Kogan; A. Stern Constraint-preserving hybrid finite element methods for Maxwell’s equations, Found. Comput. Math. (2020) | DOI

[5] F. Brezzi; J. Douglas; L. D. Marini Two families of mixed finite elements for second order elliptic problems, Numer. Math., Volume 47 (1985) no. 2, pp. 217-235 | DOI | MR | Zbl

[6] F. Brezzi; M. Fortin Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, 15, Springer, 1991, x+350 pages | MR | Zbl

[7] S. H. Christiansen; T. G. Halvorsen A simplicial gauge theory, J. Math. Phys., Volume 53 (2012) no. 3, 033501, 17 pages | DOI | MR | Zbl

[8] S. H. Christiansen; R. Winther On constraint preservation in numerical simulations of Yang-Mills equations, SIAM J. Sci. Comput., Volume 28 (2006) no. 1, pp. 75-101 | DOI | MR | Zbl

[9] S. K. Donaldson; P. B. Kronheimer The geometry of four-manifolds, Oxford Mathematical Monographs, Clarendon Press, 1990, x+440 pages (Oxford Science Publications) | MR | Zbl

[10] E. Hairer; C. Lubich; G. Wanner Geometric numerical integration, Springer Series in Computational Mathematics, 31, Springer, 2006, xviii+644 pages (Structure-preserving algorithms for ordinary differential equations) | DOI | MR

[11] R. Hiptmair Finite elements in computational electromagnetism, Acta Numer., Volume 11 (2002), pp. 237-339 | DOI | MR | Zbl

[12] A. Logg; H.-A. Mardal; G. W. Wells et al. Automated Solution of Differential Equations by the Finite Element Method, Springer, 2012 | DOI

[13] D. Mitrea; M. Mitrea; M.-C. Shaw Traces of differential forms on Lipschitz domains, the boundary de Rham complex, and Hodge decompositions, Indiana Univ. Math. J., Volume 57 (2008) no. 5, pp. 2061-2095 | DOI | MR | Zbl

[14] P. Monk Analysis of a finite element method for Maxwell’s equations, SIAM J. Numer. Anal., Volume 29 (1992) no. 3, pp. 714-729 | DOI | MR | Zbl

[15] K. K. Uhlenbeck Connections with ${L}^{p}$ bounds on curvature, Commun. Math. Phys., Volume 83 (1982) no. 1, pp. 31-42 | DOI | MR | Zbl

[16] N. Weck Traces of differential forms on Lipschitz boundaries, Analysis (Munich), Volume 24 (2004) no. 2, pp. 147-169 | DOI | MR | Zbl

[17] K. G. Wilson Confinement of quarks, Phys. Rev. D, Volume 10 (1974), pp. 2445-2459 | DOI

[18] C.-N. Yang; R. L. Mills Conservation of Isotopic Spin and Isotopic Gauge Invariance, Phys. Rev., Volume 96 (1954), pp. 191-195 | DOI | MR | Zbl

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