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

TL;DR

This work addresses the pollution effect in low-order finite element discretizations of the time-harmonic Maxwell equations at large wave numbers. By developing a stability-based, preasymptotic error analysis and a modified duality argument, the authors obtain explicit energy- and ${L^2}$-norm error bounds for the linear EEM and extend these results to the ${\boldsymbol{H}}({\boldsymbol{\mathrm{curl}}})$-conforming interior penalty EEM (CIP-EEM) with general complex penalties. The main contributions are the first preasymptotic error estimates for the second-type Nédélec EEM under the mesh constraint ${\kappa^3 h^2}$ small, together with a generalization to CIP-EEM that preserves the same error bounds while reducing pollution via carefully chosen penalties. Numerical experiments corroborate the theory and demonstrate that CIP-EEM can significantly mitigate pollution effects, offering a practical route for reliable high-frequency Maxwell simulations with low-order elements.

Abstract

Preasymptotic error estimates are derived for the linear edge element method (EEM) and the linear $\boldsymbol{H}(\boldsymbol{\mathrm{curl}})$-conforming interior penalty edge element method (CIP-EEM) for the time-harmonic Maxwell equations with large wave number. It is shown that under the mesh condition that $κ^3 h^2$ is sufficiently small, the errors of the solutions to both methods are bounded by $\mathcal{O} (κh + κ^3 h^2 )$ in the energy norm and $\mathcal{O} (κh^2 + κ^2 h^2 )$ in the $\boldsymbol{L}^2$ norm, where $κ$ is the wave number and $h$ is the mesh size. Numerical tests are provided to verify our theoretical results and to illustrate the potential of CIP-EEM in significantly reducing the pollution effect.

Figures (4)

• Figure 1: Mesh of type cub6 and penalty parameters on different interior faces.
• Figure 2: Log-log plots of the relative ${\boldsymbol{H}}(\mathop{\mathrm{\boldsymbol{\mathrm{curl}}}}\nolimits)$ errors of the EE solution (left), the CIP-EE solution (right), and the interpolation (black dotted line) versus $1/h_0$ with $1/h_0= 2, 4,\cdots, 24, 32$ and $64$, for $\kappa = 5, 20$, and $50$, respectively.
• Figure 3: Log-log plots of the relative ${\boldsymbol{L}}^2$ errors of the EE solution (left), the CIP-EE solution (right), and the interpolation (black dotted line) versus $1/h_0$ with $1/h_0= 2, 4,\cdots, 24, 32$ and $64$, for $\kappa = 5, 20$, and $50$, respectively.
• Figure 4: Relative ${\boldsymbol{H}}(\mathop{\mathrm{\boldsymbol{\mathrm{curl}}}}\nolimits)$ errors with $\kappa h_0=1$.

Theorems & Definitions (30)

• Remark 2.1
• Lemma 3.1
• proof
• Remark 3.1
• Lemma 3.2
• Lemma 3.3
• proof
• Lemma 3.4: Helmholtz Decomposition
• proof
• Theorem 4.1