Robust equilibrated flux reconstruction in h(curl) based on local minimizations: Application to a posteriori analysis of the curl-curl problem*

Abstract

We present a local construction of b 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 Nedelec's finite element approximations of the curl-curl problem. A divergence-free decomposition of a divergence-free H (div)-conforming piecewise polynomial, relying on overconstrained minimizations in Raviart-Thomas spaces, is the key ingredient. Numerical results illustrate the theoretical developments.