p-ROBUST EQUILIBRATED FLUX RECONSTRUCTION IN H(curl) BASED ON LOCAL MINIMIZATIONS: APPLICATION TO A POSTERIORI ANALYSIS OF THE CURL-CURL PROBLEM

We present a local construction of H (curl)-conforming piecewise polynomials satisfying a prescribed curl constraint. We start from a piecewise polynomial not contained in the H(curl) space but satisfying a suitable orthogonality property. The procedure employs minimizations in vertex patches, and the outcome is, up to a generic constant independent of the underlying polynomial degree, as accurate as the best approximations over the entire local versions of H(curl). This allows to design guaranteed, fully computable, constant-free, and polynomial-degree-robust a posteriori error estimates of Prager--Synge type for Ne'\de'\lec's finite element approximations of the curl-curl problem. A divergence-free decomposition of a divergence-free H(div)-conforming piece wise polynomial, relying on overconstrained minimizations in Raviart-Thomas spaces, is the key ingredient. Numerical results illustrate the theoretical developments.