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.

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.

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.

Received:
Accepted:
Published online:
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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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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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