p-robust equilibrated flux reconstruction in H(curl) based on local minimizations. Application to a posteriori analysis of the curl-curl problem

Théophile Chaumont-Frelet (1, 2) , Martin Vohralík (3, 4)

1 ATLANTIS - Modélisation et méthodes numériques pour le calcul d'interactions onde-matière nanostructurée
2 LJAD - Laboratoire Jean Alexandre Dieudonné
3 SERENA - Simulation for the Environment: Reliable and Efficient Numerical Algorithms
4 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique

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 Nédélec finite element approximations of the curl-curl problem. A divergence-free decomposition of a divergence-free H(div)-conforming piecewise polynomial, relying on over-constrained minimizations in Raviart-Thomas spaces, is the key ingredient. Numerical results illustrate the theoretical developments.

Mots clés: Sobolev space H(curl); Sobolev space H(div); Equilibrated flux reconstruction; p-robustness; A posteriori error estimate; Divergence-free decomposition; Broken polynomial extension

Domaines: Analyse numérique [math.NA]

Théophile Chaumont-Frelet, Martin Vohralík. p-robust equilibrated flux reconstruction in H(curl) based on local minimizations. Application to a posteriori analysis of the curl-curl problem. 2021. ⟨hal-03227570v1⟩