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

TL;DR

This paper proves that discrete minimizers in Nédélec polynomial spaces on a tetrahedron are uniformly stable with respect to polynomial degree p, which is crucial for robust error estimation in Maxwell equations.

Contribution

It establishes polynomial-degree-robust stability of discrete minimizers in H(curl) on tetrahedra, extending previous theoretical results to practical finite element analysis.

Findings

Discrete minimizers perform as well as continuous ones independent of polynomial degree p.

The stability result supports the development of robust a posteriori error estimators.

Applicable to Maxwell equations in various regimes, improving numerical reliability.

Abstract

We prove that the minimizer in the N\'ed\'elec 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. Sch\"oberl 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.