Polynomial preserving recoveries of edge element method on Cartesian grids for the time-harmonic Maxwell equations with large wave number
Abstract
This paper considers the lowest-order first type Nédélec edge element method (EEM) on Cartesian grids for the three-dimensional time-harmonic Maxwell equations with a large wave number. New polynomial preserving recovery (PPR) operators are proposed for the curl of the edge element solution and for the solution itself, respectively. Under the condition that κ3 h2 Csol is sufficiently small, second-order superconvergence estimates are proved for both the recovered curl and the recovered solution, where κ is the wave number, h is the mesh size, and Csol is a stability constant associated with the Maxwell solution operator. In particular, the analysis shows that the proposed PPR procedures cannot mitigate the well-known pollution effect inherent to the EEM. To reduce the pollution error, we further propose a new continuous interior penalty edge element method (CIP-EEM) that incorporates an additional normal-jump penalty term. It is shown that by appropriately choosing the penalty parameters, the new CIP-EEM can improve the phase error by two orders in κh. Numerical experiments are presented to confirm the theoretical superconvergence results and to demonstrate that the CIP-EEM can effectively reduce the pollution error in the high-frequency regime.
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BibTeX
@misc{lu2026polynomial,
title = {{Polynomial preserving recoveries of edge element method on Cartesian grids for the time-harmonic Maxwell equations with large wave number}},
author = {Shuaishuai Lu and Haijun Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11064},
eprint = {2609.11064},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11064},
}
RIS
TY - UNPB
TI - Polynomial preserving recoveries of edge element method on Cartesian grids for the time-harmonic Maxwell equations with large wave number
AU - Shuaishuai Lu
AU - Haijun Wu
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11064
UR - https://arxiv.org/abs/2609.11064
AB - This paper considers the lowest-order first type Nédélec edge element method (EEM) on Cartesian grids for the three-dimensional time-harmonic Maxwell equations with a large wave number. New polynomial preserving recovery (PPR) operators are proposed for the curl of the edge element solution and for the solution itself, respectively. Under the condition that κ^3 h^2 C_sol is sufficiently small, second-order superconvergence estimates are proved for both the recovered curl and the recovered solution, where κ is the wave number, h is the mesh size, and C_sol is a stability constant associated with the Maxwell solution operator. In particular, the analysis shows that the proposed PPR procedures cannot mitigate the well-known pollution effect inherent to the EEM. To reduce the pollution error, we further propose a new continuous interior penalty edge element method (CIP-EEM) that incorporates an additional normal-jump penalty term. It is shown that by appropriately choosing the penalty parameters, the new CIP-EEM can improve the phase error by two orders in κh. Numerical experiments are presented to confirm the theoretical superconvergence results and to demonstrate that the CIP-EEM can effectively reduce the pollution error in the high-frequency regime.
ER -