Asymptotic optimality of the edge finite element approximation of the time-harmonic Maxwell’s equations

Théophile Chaumont-Frelet; Alexandre Ern

doi:10.5802/crmath.757

Article de recherche - Analyse numérique, Équations aux dérivées partielles

Asymptotic optimality of the edge finite element approximation of the time-harmonic Maxwell’s equations
[Optimalité asymptotique pour l’approximation par éléments finis de Nédélec des équations de Maxwell en régime harmonique]

Théophile Chaumont-Frelet ¹ ; Alexandre Ern ^2,³

¹ Inria Univ. Lille and Laboratoire Paul Painlevé, 59655 Villeneuve-d’Ascq, France
² CERMICS, ENPC, Institut Polytechnique de Paris, 77455 Marne-la-Vallée cedex 2, France
³ Inria, 48 rue Barrault, 75647 Paris, France

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1083-1101

Abstract (VO)
Résumé

We analyze the conforming approximation of the time-harmonic Maxwell’s equations using Nédélec (edge) finite elements. We prove that the approximation is asymptotically optimal, i.e., the approximation error in the energy norm is bounded by the best-approximation error times a constant that tends to one as the mesh is refined and/or the polynomial degree is increased. Moreover, under the same conditions on the mesh and/or the polynomial degree, we establish discrete inf-sup stability with a constant that corresponds to the continuous constant up to a factor of two at most. Our proofs apply under minimal regularity assumptions on the exact solution, so that general domains, material coefficients, and right-hand sides are allowed.

On analyse l’approximation conforme des équations de Maxwell en régime harmonique par des éléments finis de Nédélec. On montre l’optimalité asymptotique de l’approximation, i.e., l’erreur d’approximation en norme d’énergie est bornée par l’erreur de la meilleure approximation multipliée par une constante qui tend vers un quand le pas du maillage tend vers zéro ou le degré polynomial tend vers l’infini. De plus, sous ces mêmes hypothèses sur le maillage et le degré polynomial, on établit une condition de stabilité inf-sup avec une constante qui vaut au plus deux fois la valeur de la constante de stabilité du problème continu. Les preuves s’appliquent sous des hypothèses de régularité minimale sur la solution, ce qui permet de considérer une large classe de domaines, propriétés matériaux et termes sources.

Reçu le : 2024-05-24
Révisé le : 2025-02-18
Accepté le : 2025-05-06
Publié le : 2025-10-13

DOI : 10.5802/crmath.757

Classification : 65N30, 78M10, 65N15
Keywords: Electromagnetics, finite element methods, Maxwell’s equations, duality argument, asymptotic optimality
Mots-clés : Électromagnétisme, méthodes d’éléments finis, équations de Maxwell, argument de dualité, optimalité asymptotique

Affiliations des auteurs :

Théophile Chaumont-Frelet ¹ ; Alexandre Ern ^{2, 3}

¹ Inria Univ. Lille and Laboratoire Paul Painlevé, 59655 Villeneuve-d’Ascq, France
² CERMICS, ENPC, Institut Polytechnique de Paris, 77455 Marne-la-Vallée cedex 2, France
³ Inria, 48 rue Barrault, 75647 Paris, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Th\'eophile Chaumont-Frelet and Alexandre Ern},
     title = {Asymptotic optimality of the edge finite element approximation of the time-harmonic {Maxwell{\textquoteright}s} equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1083--1101},
     year = {2025},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     doi = {10.5802/crmath.757},
     language = {en},
}

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Théophile Chaumont-Frelet; Alexandre Ern. Asymptotic optimality of the edge finite element approximation of the time-harmonic Maxwell’s equations. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1083-1101. doi: 10.5802/crmath.757

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