Comptes Rendus
Optimisation
Polynomial-degree-robust H(curl)-stability of discrete minimization in a tetrahedron
Comptes Rendus. Mathématique, Volume 358 (2020) no. 9-10, pp. 1101-1110.

On prouve que le minimiseur dans l’espace des polynômes de Nédélec d’un certain degré p0 d’un problème de minimisation discret est aussi efficace que le minimiseur dans tout H(curl), à une constante indépendante de p près. Les problèmes de minimisation considérés concernent des champs de vecteurs définis sur un tétraèdre non dégénéré de 3 avec des contraintes polynomiales imposées sur le rotationnel et sur la restriction de la trace tangentielle à certaines faces du tétraèdre. Ce résultat, basé sur [L. Demkowicz, J. Gopalakrishnan, J. Schöberl, SIAM J. Numer. Anal. 47 (2009), 3293–3324] et [M. Costabel, A. McIntosh, Math. Z. 265 (2010), 297–320], est un outil fondamental pour construire des estimateurs a posteriori robustes vis à vis du degré p dans le contexte de l’approximation des équations de Maxwell.

We prove that the minimizer in the Nédélec polynomial space of some degree p0 of a discrete minimization problem performs as well as the continuous minimizer in H(curl), up to a constant that is independent of the polynomial degree p. The minimization problems are posed for fields defined on a single non-degenerate tetrahedron in 3 with polynomial constraints enforced on the curl of the field and its tangential trace on some faces of the tetrahedron. This result builds upon [L. Demkowicz, J. Gopalakrishnan, J. Schöberl, SIAM J. Numer. Anal. 47 (2009), 3293–3324] and [M. Costabel, A. McIntosh, Math. Z. 265 (2010), 297–320] and is a fundamental ingredient to build polynomial-degree-robust a posteriori error estimators when approximating the Maxwell equations in several regimes leading to a curl-curl problem.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.133
Classification : 65N15, 65N30, 76M10

Théophile Chaumont-Frelet 1, 2 ; Alexandre Ern 3, 4 ; Martin Vohralík 3, 4

1 Inria, 2004 Route des Lucioles, 06902 Valbonne, France
2 Laboratoire J.A. Dieudonné, Parc Valrose, 28 Avenue Valrose, 06000 Nice, France
3 CERMICS, École des Ponts, 77455 Marne-la-Vallée cedex 2, France
4 Inria, 2 rue Simone Iff, 75589 Paris, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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     title = {Polynomial-degree-robust $\protect H({\protect \bf curl})$-stability of discrete minimization in a tetrahedron},
     journal = {Comptes Rendus. Math\'ematique},
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Théophile Chaumont-Frelet; Alexandre Ern; Martin Vohralík. Polynomial-degree-robust $\protect H({\protect \bf curl})$-stability of discrete minimization in a tetrahedron. Comptes Rendus. Mathématique, Volume 358 (2020) no. 9-10, pp. 1101-1110. doi : 10.5802/crmath.133. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.133/

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