We prove that the minimizer in the Nédélec polynomial space of some degree of a discrete minimization problem performs as well as the continuous minimizer in , up to a constant that is independent of the polynomial degree . The minimization problems are posed for fields defined on a single non-degenerate tetrahedron in 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 d’un problème de minimisation discret est aussi efficace que le minimiseur dans tout , à une constante indépendante de près. Les problèmes de minimisation considérés concernent des champs de vecteurs définis sur un tétraèdre non dégénéré de 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 dans le contexte de l’approximation des équations de Maxwell.
Accepted:
Published online:
Théophile Chaumont-Frelet 1, 2; Alexandre Ern 3, 4; Martin Vohralík 3, 4
@article{CRMATH_2020__358_9-10_1101_0, 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}, }
TY - JOUR AU - Théophile Chaumont-Frelet AU - Alexandre Ern AU - Martin Vohralík TI - Polynomial-degree-robust $\protect H({\protect \bf curl})$-stability of discrete minimization in a tetrahedron JO - Comptes Rendus. Mathématique PY - 2020 SP - 1101 EP - 1110 VL - 358 IS - 9-10 PB - Académie des sciences, Paris DO - 10.5802/crmath.133 LA - en ID - CRMATH_2020__358_9-10_1101_0 ER -
%0 Journal Article %A Théophile Chaumont-Frelet %A Alexandre Ern %A Martin Vohralík %T Polynomial-degree-robust $\protect H({\protect \bf curl})$-stability of discrete minimization in a tetrahedron %J Comptes Rendus. Mathématique %D 2020 %P 1101-1110 %V 358 %N 9-10 %I Académie des sciences, Paris %R 10.5802/crmath.133 %G en %F CRMATH_2020__358_9-10_1101_0
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/
[1] Equilibrated residual error estimates are -robust, Comput. Methods Appl. Mech. Eng., Volume 198 (2009) no. 13-14, pp. 1189-1197 | DOI | MR | Zbl
[2] Stable broken polynomial extensions and -robust quasi-equilibrated a posteriori estimators for Maxwell’s equations (2020) (https://hal.inria.fr/hal-02644173, submitted for publication)
[3] 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] An adaptive -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] Polynomial extension operators. Part I, SIAM J. Numer. Anal., Volume 46 (2008) no. 6, pp. 3006-3031 | DOI | MR | Zbl
[6] Polynomial extension operators. Part II, SIAM J. Numer. Anal., Volume 47 (2009) no. 5, pp. 3293-3324 | DOI | MR | Zbl
[7] Polynomial extension operators. Part III, Math. Comp., Volume 81 (2012) no. 279, pp. 1289-1326 | DOI | MR
[8] Finite Elements I: Approximation and interpolation, Springer, 2020 (In press) | Zbl
[9] 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] Stable broken and 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] 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] Mixed finite elements in , Numer. Math., Volume 35 (1980) no. 3, pp. 315-341 | DOI | MR | Zbl
[13] 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
Cited by Sources:
Comments - Policy