Polynomial-degree-robust $\protect H({\protect \bf curl})$-stability of discrete minimization in a tetrahedron

Théophile Chaumont-Frelet; Alexandre Ern; Martin Vohralík

doi:10.5802/crmath.133

Optimisation

Polynomial-degree-robust

H (curl)

-stability of discrete minimization in a tetrahedron

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

Comptes Rendus. Mathématique, Volume 358 (2020) no. 9-10, pp. 1101-1110.

Abstract (VO)
Résumé

We prove that the minimizer in the Nédélec polynomial space of some degree $p \geq 0$ 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 $ℝ^{3}$ 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 $p \geq 0$ 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 $ℝ^{3}$ 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.

Reçu le : 2020-05-27
Accepté le : 2020-10-17
Publié le : 2021-01-05

DOI : 10.5802/crmath.133

Classification : 65N15, 65N30, 76M10

Affiliations des auteurs :

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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     author = {Th\'eophile Chaumont-Frelet and Alexandre Ern and Martin Vohral{\'\i}k},
     title = {Polynomial-degree-robust $\protect H({\protect \bf curl})$-stability of discrete minimization in a tetrahedron},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1101--1110},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {9-10},
     year = {2020},
     doi = {10.5802/crmath.133},
     language = {en},
}

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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/

Bibliographie
Cité par

[1] Dietrich Braess; Veronika Pillwein; Joachim Schöberl Equilibrated residual error estimates are $p$ -robust, Comput. Methods Appl. Mech. Eng., Volume 198 (2009) no. 13-14, pp. 1189-1197 | DOI | MR | Zbl

[2] Théophile Chaumont-Frelet; Alexandre Ern; Martin Vohralík Stable broken $H (curl)$ polynomial extensions and $p$ -robust quasi-equilibrated a posteriori estimators for Maxwell’s equations (2020) (https://hal.inria.fr/hal-02644173, submitted for publication)

[3] Martin Costabel; Alan McIntosh On Bogovskiĭand regularized Poincaré integral operators for de Rham complexes on Lipschitz domains, Math. Z., Volume 265 (2010) no. 2, pp. 297-320 | DOI | MR | Zbl

[4] Patrik Daniel; Alexandre Ern; Iain Smears; Martin Vohralík An adaptive $h p$ -refinement strategy with computable guaranteed bound on the error reduction factor, Comput. Math. Appl., Volume 76 (2018) no. 5, pp. 967-983 | DOI | MR | Zbl

[5] Leszek Demkowicz; Jayadeep Gopalakrishnan; Joachim Schöberl Polynomial extension operators. Part I, SIAM J. Numer. Anal., Volume 46 (2008) no. 6, pp. 3006-3031 | DOI | MR | Zbl

[6] Leszek Demkowicz; Jayadeep Gopalakrishnan; Joachim Schöberl Polynomial extension operators. Part II, SIAM J. Numer. Anal., Volume 47 (2009) no. 5, pp. 3293-3324 | DOI | MR | Zbl

[7] Leszek Demkowicz; Jayadeep Gopalakrishnan; Joachim Schöberl Polynomial extension operators. Part III, Math. Comp., Volume 81 (2012) no. 279, pp. 1289-1326 | DOI | MR

[8] Alexandre Ern; Jean-Luc Guermond Finite Elements I: Approximation and interpolation, Springer, 2020 (In press) | Zbl

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[10] Alexandre Ern; Martin Vohralík Stable broken $H^{1}$ and $H (div)$ polynomial extensions for polynomial-degree-robust potential and flux reconstruction in three space dimensions, Math. Comp., Volume 89 (2020) no. 322, pp. 551-594 | DOI | MR | Zbl

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