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

TL;DR

This paper establishes stability estimates for acoustic and electromagnetic waveguides in a time-harmonic setting, and introduces a novel UW DPG method with non-polynomial functions to effectively handle long waveguides.

Contribution

It provides the first stability estimates with linear dependence on waveguide length and develops a new UW DPG method for long waveguides using non-polynomial ansatz functions.

Findings

Stability constants depend linearly on waveguide length.

Scaling the test norm mitigates stability deterioration for long waveguides.

The full envelope approximation enables treatment of long waveguides with UW DPG.

Abstract

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