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

TL;DR

This paper derives preasymptotic error estimates for EEM and CIP-EEM methods applied to time-harmonic Maxwell equations with large wave numbers, showing how errors depend on mesh size and wave number.

Contribution

It provides new error bounds under specific mesh conditions for EEM and CIP-EEM, highlighting the effectiveness of CIP-EEM in reducing pollution effects.

Findings

Error bounds depend on wave number and mesh size

CIP-EEM reduces pollution effect

Numerical tests confirm theoretical estimates

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 $\kappa^3 h^2$ is sufficiently small, the errors of the solutions to both methods are bounded by $\mathcal{O} (\kappa h + \kappa^3 h^2 )$ in the energy norm and $\mathcal{O} (\kappa h^2 + \kappa^2 h^2 )$ in the $\boldsymbol{L}^2$ norm, where $\kappa$ 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.