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Polynomial-degree-robust H(curl)-stability of discrete minimization in a tetrahedron

Abstract : We prove that the minimizer in the Nédélec polynomial space of some degree p ≥ 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.
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Submitted on : Wednesday, May 27, 2020 - 7:49:54 AM
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Théophile Chaumont-Frelet, Alexandre Ern, Martin Vohralík. Polynomial-degree-robust H(curl)-stability of discrete minimization in a tetrahedron. Comptes Rendus Mathématique, Elsevier Masson, 2020, 358 (9-10), pp.1101-1110. ⟨10.5802/crmath.133⟩. ⟨hal-02631319⟩

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