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