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}

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},
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     year = {2020},
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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

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[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

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Théophile Chaumont-Frelet; Martin Vohralík Constrained and Unconstrained Stable Discrete Minimizations for p-Robust Local Reconstructions in Vertex Patches in the de Rham Complex, Foundations of Computational Mathematics (2024) | DOI:10.1007/s10208-024-09674-7
Carsten Carstensen; Benedikt Gräßle; Ngoc Tien Tran Adaptive hybrid high-order method for guaranteed lower eigenvalue bounds, Numerische Mathematik, Volume 156 (2024) no. 3, pp. 813-851 | DOI:10.1007/s00211-024-01407-w | Zbl:1548.65259
Théophile Chaumont-Frelet; Martin Vohralík A stable local commuting projector and optimal $h p$ approximation estimates in $H (curl)$ , Numerische Mathematik, Volume 156 (2024) no. 6, pp. 2293-2342 | DOI:10.1007/s00211-024-01431-w | Zbl:7949945
T. Chaumont-Frelet A simple equilibration procedure leading to polynomial-degree-robust a posteriori error estimators for the curl-curl problem, Mathematics of Computation, Volume 92 (2023) no. 344, pp. 2413-2437 | DOI:10.1090/mcom/3817 | Zbl:1522.65202
Théophile Chaumont-Frelet; Martin Vohralík $p$ -robust equilibrated flux reconstruction in $H (curl)$ based on local minimizations: application to a posteriori analysis of the curl-curl problem, SIAM Journal on Numerical Analysis, Volume 61 (2023) no. 4, pp. 1783-1818 | DOI:10.1137/21m141909x | Zbl:1535.65275
T. Chaumont-Frelet; A. Ern; M. Vohralík Stable broken $H (curl)$ polynomial extensions and $p$ -robust a posteriori error estimates by broken patchwise equilibration for the curl-curl problem, Mathematics of Computation, Volume 91 (2022) no. 333, pp. 37-74 | DOI:10.1090/mcom/3673 | Zbl:1479.65027
Théophile Chaumont-Frelet; Martin Vohralík Equivalence of local-best and global-best approximations in $H (curl)$ , Calcolo, Volume 58 (2021) no. 4, p. 12 (Id/No 53) | DOI:10.1007/s10092-021-00430-9 | Zbl:1490.65264
Joscha Gedicke; Sjoerd Geevers; Ilaria Perugia; Joachim Schöberl A polynomial-degree-robust a posteriori error estimator for Nédélec discretizations of magnetostatic problems, SIAM Journal on Numerical Analysis, Volume 59 (2021) no. 4, pp. 2237-2253 | DOI:10.1137/20m1333365 | Zbl:1492.65307

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