Length-explicit stability analysis of Helmholtz problems in leaky circular waveguides
Abstract
Motivated by the study and simulation of long, coiled optical fibers we consider in this article a simplified model that is prevalent in the engineering community. Mathematically, the problem is specified as follows: Time-harmonic wave propagation is modeled by the Helmholtz equation; the waveguide is a bounded circular section with a transparent boundary condition on one end; the dissipation of energy is modeled by an impedance boundary condition on the outer hull of the waveguide. We show a stability estimate that is explicit in terms of the angular length of the waveguide. The analysis is based on a separation of variables ansatz and the study of the related (nonselfadjoint) modal eigenvalue problem. The key property there is to show that the modes form a Riesz basis in both and spaces. To this end we apply perturbation theory for selfadjoint operators and the concept of local subordination of perturbations [B. Mityagin and P. Siegl, JAM 139 (2019)]. Since the possibility of nontrivial Jordan chains cannot be ruled out, our whole methodology is conducted accordingly. In addition, in contrast to previous works, we include a bounded but heterogeneous part of the waveguide into our considered setting.
Cite
@article{arxiv.2603.15107,
title = {Length-explicit stability analysis of Helmholtz problems in leaky circular waveguides},
author = {Leszek Demkowicz and Martin Halla and Jens Markus Melenk},
journal= {arXiv preprint arXiv:2603.15107},
year = {2026}
}