Symmetry Inheritance and Exact Mirror Symmetry Reduction for Electric Field Vector Wave Problems with PEC Boundaries
Abstract
A symmetry-inheritance and exact domain-reduction framework is developed for the second-order vector wave equation of the electric field in bounded domains with a perfect electric conductor (PEC) boundary and general spatially varying anisotropic media. Orthogonal group actions on polar vectors, axial vectors, and second-order constitutive tensors are used to derive the transformed curl--curl equation and PEC trace directly from the original boundary-value problem. Under unique solvability, the common symmetry group of the problem data is inherited by the electric field. When the symmetry group of the electromagnetic medium admits a mirror symmetry, an arbitrary impressed current can be decomposed by the associated reflection projectors into mirror-even and mirror-odd components. The corresponding full-domain fields inherit the respective parities, and their exact restriction to one half-domain yields a perfect magnetic conductor (PMC) condition for the even field and a PEC condition for the odd field. The original problem is therefore represented exactly by two independent reduced-domain problems. A symmetry-compatible first-family Nédélec discretization preserves this decomposition at the discrete level. Analytical and numerical results verify the symmetry-inheritance theory, the exact PEC/PMC reduced-domain formulations, and the discrete reconstruction for constant and spatially varying anisotropic conductive media under both symmetric and asymmetric excitations.
Related
- Symmetry Inheritance and Symmetry-Reduced Finite Element Analysis for Second-Order Linear Elliptic Problems with Robin Boundary Conditions
- A 2+1-dimensional extended Dym equation: moving boundary problems solvable via Painlevé II symmetry reduction
- An Extended 2+1-Dimensional Gardner Equation: On Moving Boundary Problems Solvable via Ermakov-Painlevé II Symmetry Reduction
- Wasserstein Stability and Free Boundaries in Measure-Parameterized Bilevel Obstacle Problems
- IQFEM: A quantum finite element method for heterogeneous problems with immersed boundaries
BibTeX
@misc{jiang2026symmetry,
title = {{Symmetry Inheritance and Exact Mirror Symmetry Reduction for Electric Field Vector Wave Problems with PEC Boundaries}},
author = {Wei Jiang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29481},
eprint = {2610.29481},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29481},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Symmetry Inheritance and Exact Mirror Symmetry Reduction for Electric Field Vector Wave Problems with PEC Boundaries
AU - Wei Jiang
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29481
UR - https://arxiv.org/abs/2610.29481
AB - A symmetry-inheritance and exact domain-reduction framework is developed for the second-order vector wave equation of the electric field in bounded domains with a perfect electric conductor (PEC) boundary and general spatially varying anisotropic media. Orthogonal group actions on polar vectors, axial vectors, and second-order constitutive tensors are used to derive the transformed curl--curl equation and PEC trace directly from the original boundary-value problem. Under unique solvability, the common symmetry group of the problem data is inherited by the electric field. When the symmetry group of the electromagnetic medium admits a mirror symmetry, an arbitrary impressed current can be decomposed by the associated reflection projectors into mirror-even and mirror-odd components. The corresponding full-domain fields inherit the respective parities, and their exact restriction to one half-domain yields a perfect magnetic conductor (PMC) condition for the even field and a PEC condition for the odd field. The original problem is therefore represented exactly by two independent reduced-domain problems. A symmetry-compatible first-family Nédélec discretization preserves this decomposition at the discrete level. Analytical and numerical results verify the symmetry-inheritance theory, the exact PEC/PMC reduced-domain formulations, and the discrete reconstruction for constant and spatially varying anisotropic conductive media under both symmetric and asymmetric excitations.
ER -