Preasymptotic error estimates of higher-order EEM for the time-harmonic Maxwell equations with large wave number

TL;DR

This work establishes preasymptotic error estimates for the time-harmonic Maxwell equations with impedance boundary condition when discretized by the second-type Nédélec edge element method of order $p$. By constructing a Maxwell-specific elliptic operator via a explicit curl-curl eigenfunction truncation and employing a modified duality argument, the authors prove κ-explicit stability and derive energy- and ${oldsymbol{L}}^2$-error bounds under the mesh condition $oldsymbol{κ}^{2p+1} h^{2p}$ being small. The main results show the EEM error in the energy norm scales as $igO(oldsymbol{κ}^{p} h^{p} + oldsymbol{κ}^{2p+1} h^{2p})$ and the ${oldsymbol{L}}^2$-type error scales as $igO((oldsymbol{κ} h)^{p+1} + oldsymbol{κ}^{2p+1} h^{2p})$, with κ-dependent constants absent from the final bounds. The analysis blends wave-number-aware regularity decompositions, discrete Helmholtz decompositions, elliptic projections, and the truncation-based smoothing to extend preasymptotic error theory from Helmholtz to Maxwell problems, complemented by numerical tests that illustrate the predicted behavior and the benefits of higher-order EEM in the preasymptotic regime.

Abstract

The time-harmonic Maxwell equations with impedance boundary condition and large wave number are discretized using the second-type Nédélec's edge element method (EEM). Preasymptotic error bounds are derived, showing that, under the mesh condition $κ^{2p+1}h^{2p}$ being sufficiently small, the error of the EEM of order $p$ in the energy norm is bounded by $\mathcal{O}\big(κ^{p}h^p + κ^{2p+1}h^{2p}\big)$, while the error in the $κ$-scaled $\boldsymbol{L}^2$ norm is bounded by $\mathcal{O}\big((κh)^{p+1} + κ^{2p+1} h^{2p}\big)$. Here, $κ$ is the wave number and $h$ is the mesh size. Numerical tests are provided to illustrate our theoretical results.

Figures (3)

• Figure 1: A sample mesh and relative error curve.
• Figure 2: Log-log plots of the relative errors in $\interleave\cdot\interleave_{E}$ of the EE solution versus $N_{\lambda}$ with $p= 1,2,3$, and for $\kappa = 5$ and $50$, respectively.
• Figure 3: Log-log plots of the relative ${\boldsymbol{L}}^2$ errors of the EE solution versus $N_{\lambda}$ with $p= 1,2,3$, and for $\kappa = 5$ and $50$, respectively.

Theorems & Definitions (37)

• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3: Helmholtz decomposition of ${\boldsymbol{L}}^2(\Omega)$
• Remark 2.1
• Lemma 2.4: Helmholtz decompositions of ${\boldsymbol{V}_{h}^{0}}$
• proof
• Lemma 2.5
• Theorem 2.1
• proof