Trefftz Discontinuous Galerkin approximation of an acoustic waveguide

We propose a modified Trefftz Discontinuous Galerkin (TDG) method for approximating a time-harmonic acoustic scattering problem in an infinitely elongated waveguide. In the waveguide we suppose there is a bounded, penetrable and possibly absorbing scatterer. The classical TDG is not applicable to this important case. Novel features of our modified TDG method are that it is applicable when the scatterer is absorbing, and it uses a stable treatment of the asymptotic radiation condition for the scattered field. For the modified TDG, we prove $h$ and $p$-convergence in the $L^2$ norm. The theoretical results are verified numerically for a discretization based on plane waves (that may be evanescent in the scatterer).