Asymptotically constant-free and polynomial-degree-robust a posteriori error estimates for time-harmonic Maxwell's equations

TL;DR

The paper develops an equilibrated, a posteriori error estimator for time-harmonic Maxwell equations discretized with Nédélec elements. It introduces fully local reconstructions of the electric displacement and magnetic field on vertex patches to enforce divergence and curl constraints, enabling a Prager-Synge type upper bound and a p-robust, low-frequency robust lower bound. The estimator η_K is shown to be reliable with a constant tending to one under refinement and/or increasing polynomial degree, while remaining efficient and insensitive to p in the lower bound; numerical tests corroborate the theory across continuous and discrete settings. The work advances robust adaptive procedures for Maxwell problems by connecting equilibrated estimators to local reconstructions that respect the underlying physics, including topology and divergence constraints.

Abstract

We propose a novel a posteriori error estimator for the Nédélec finite element discretization of time-harmonic Maxwell's equations. After the approximation of the electric field is computed, we propose a fully localized algorithm to reconstruct approximations to the electric displacement and the magnetic field, with such approximations respectively fulfilling suitable divergence and curl constraints. These reconstructed fields are in turn used to construct an a posteriori error estimator which is shown to be reliable and efficient. Specifically, the estimator controls the error from above up to a constant that tends to one as the mesh is refined and/or the polynomial degree is increased, and from below up to constant independent of $p$. Both bounds are also fully-robust in the low-frequency regime. The properties of the proposed estimator are illustrated on a set of numerical examples.

Figures (8)

• Figure 6.1: $m=3$ and $p=1$
• Figure 6.2: $m=3$ and $p=2$
• Figure 6.3: $m=3$ and $p=3$
• Figure 6.4: Effectivity indices $\mathcal{I}$ for the case $m=3$
• Figure 6.5: $m=5$ and $p=1$

Theorems & Definitions (26)

• Theorem 3.1: Error estimates in energy norm
• Theorem 3.2: Equilibrated local reconstructions
• Theorem 3.3: Efficiency
• Lemma 4.1: Residual term
• proof
• Lemma 4.2: Error estimates in $\boldsymbol L^2(\Omega)$ norm
• proof
• proof : Proof of Theorem \ref{['theorem_reliability']}
• Lemma 5.1: Idealized electric displacement reconstruction
• proof