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.

Résumé
Abstract

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.

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.

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}

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

[9] Alexandre Ern; Martin Vohralík Polynomial-degree-robust a posteriori estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed discretizations, SIAM J. Numer. Anal., Volume 53 (2015) no. 2, pp. 1058-1081 | DOI | MR | Zbl

[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

[11] Paolo Fernandes; Gianni Gilardi Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions, Math. Models Methods Appl. Sci., Volume 7 (1997) no. 7, pp. 957-991 | DOI | MR | Zbl

[12] Jean-Claude Nédélec Mixed finite elements in $ℝ^{3}$ , Numer. Math., Volume 35 (1980) no. 3, pp. 315-341 | DOI | MR | Zbl

[13] Pierre-Arnaud Raviart; Jean-Marie Thomas A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) (Lecture Notes in Mathematics), Volume 606, Springer, 1977, pp. 292-315 | MR

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