Polynomial-degree-robust $H\left(\mathbf{curl}\right)$-stability of discrete minimization in a tetrahedron

Chaumont-Frelet, Théophile; Ern, Alexandre; Vohralík, Martin

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

On prouve que le minimiseur dans l’espace des polynômes de Nédélec d’un certain degré $p\ge 0$ d’un problème de minimisation discret est aussi efficace que le minimiseur dans tout $H\left(\mathbf{curl}\right)$, 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 $p\ge 0$ of a discrete minimization problem performs as well as the continuous minimizer in $H\left(\mathbf{curl}\right)$, 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.

Classification : 65N15, 65N30, 76M10

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