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arXiv · 2603.15107

Length-explicit stability analysis of Helmholtz problems in leaky circular waveguides

Leszek Demkowicz·Martin Halla·Jens Markus Melenk
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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 $L^2$ and $H^1$ 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.

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Leszek Demkowicz, Martin Halla, Jens Markus Melenk. 2026-03-16. Length-explicit stability analysis of Helmholtz problems in leaky circular waveguides. https://arxiv.org/abs/2603.15107

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