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

Abstract

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 $H({\bf curl})$, 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 $\mathbb{R}^3$ 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.