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In a time-harmonic setting, we show for heterogeneous acoustic andhomogeneous electromagnetic wavesguides stability estimates with the stabilityconstant depending linearly on the length $L$ of the waveguide. These stabilityestimates are used for the analysis of the (ideal) ultraweak (UW) variant ofthe Discontinuous Petrov Galerkin (DPG) method. For this UW DPG, we show thatthe stability deterioration with $L$ can be countered by suitably scaling thetest norm of the method. We present the ``full envelope approximation'', a UWDPG method based on non-polynomial ansatz functions that allows for treatinglong waveguides.

Stability Analysis for Electromagnetic Waveguides. Part 1: Acoustic and Homogeneous Electromagnetic Waveguides