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

TL;DR

This paper introduces a local polynomial reconstruction method for H(curl) spaces that enables reliable, degree-robust a posteriori error estimation for curl-curl problems, enhancing finite element analysis accuracy.

Contribution

It proposes a novel local minimization-based construction of H(curl)-conforming polynomials that leads to guaranteed, fully computable error estimates independent of polynomial degree.

Findings

The method achieves accuracy comparable to the best local approximations.

It provides guaranteed, constant-free a posteriori error bounds.

Numerical tests confirm the theoretical robustness and effectiveness.

Abstract

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 N\'ed\'elec 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.