Quasi-Trefftz spaces for the first-order time-harmonic Maxwell's equations
Abstract
The goal of this article is to study optimal high-order Taylor-based polynomial quasi-Trefftz spaces for first-order time-harmonic Maxwell's equations. Compared with standard polynomial spaces, the purpose of quasi-Trefftz spaces is to retain the same high-order best approximation property but more efficiently: their dimension is considerably smaller. To achieve this goal, they approximate only smooth solutions to the governing equation rather than to approximate general smooth functions. While there is a natural definition of Taylor-based polynomial quasi-Trefftz spaces for scalar equations, the situation is fundamentally different for vector-valued equations, as the naive quasi-Trefftz property does not systematically define the smallest space with the desired best approximation property. This article identifies the optimal quasi-Trefftz spaces for first-order time-harmonic Maxwell's equations, proposes two approaches to construct quasi-Trefftz bases, and proves the spaces' best approximation property. The theoretical results, which strongly leverage polynomial Helmholtz decompositions and the exactness of the polynomial de Rham sequences, are validated with numerical experiments.
Related
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- Polynomial preserving recoveries of edge element method on Cartesian grids for the time-harmonic Maxwell equations with large wave number
- Efficient C1 Bernstein Quasi-Trefftz Discretization for Heterogeneous High-Frequency Helmholtz Problems
- Optimization-based numerical methods for the two-and three-dimensional time harmonic Maxwell problem with sign-changing coefficients
BibTeX
@misc{fontana2026quasitrefftz,
title = {{Quasi-Trefftz spaces for the first-order time-harmonic Maxwell's equations}},
author = {Ilaria Fontana and Lise-Marie Imbert-Gérard},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05975},
eprint = {2610.05975},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05975},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Quasi-Trefftz spaces for the first-order time-harmonic Maxwell's equations
AU - Ilaria Fontana
AU - Lise-Marie Imbert-Gérard
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.05975
UR - https://arxiv.org/abs/2610.05975
AB - The goal of this article is to study optimal high-order Taylor-based polynomial quasi-Trefftz spaces for first-order time-harmonic Maxwell's equations. Compared with standard polynomial spaces, the purpose of quasi-Trefftz spaces is to retain the same high-order best approximation property but more efficiently: their dimension is considerably smaller. To achieve this goal, they approximate only smooth solutions to the governing equation rather than to approximate general smooth functions. While there is a natural definition of Taylor-based polynomial quasi-Trefftz spaces for scalar equations, the situation is fundamentally different for vector-valued equations, as the naive quasi-Trefftz property does not systematically define the smallest space with the desired best approximation property. This article identifies the optimal quasi-Trefftz spaces for first-order time-harmonic Maxwell's equations, proposes two approaches to construct quasi-Trefftz bases, and proves the spaces' best approximation property. The theoretical results, which strongly leverage polynomial Helmholtz decompositions and the exactness of the polynomial de Rham sequences, are validated with numerical experiments.
ER -