A Trefftz Discontinuous Galerkin Method for Time Harmonic Waves with Generalized Impedance Boundary Conditions
For researchers in computational wave propagation, this provides a convergent numerical scheme for a class of boundary conditions that were previously challenging to handle with TDG methods.
The paper extends a Trefftz Discontinuous Galerkin method to handle generalized impedance boundary conditions (GIBCs) arising in scattering problems with thin coatings and higher-order absorbing boundary conditions. Convergence is proved and demonstrated with two numerical examples.
We show how a Trefftz Discontinuous Galerkin (TDG) method for the displacement form of the Helmholtz equation can be used to approximate problems having a generalized impedance boundary condition (GIBC) involving surface derivatives of the solution. Such boundary conditions arise naturally when modeling scattering from a scatterer with a thin coating. The thin coating can then be approximated by a GIBC. A second place GIBCs arise is as higher order absorbing boundary conditions. This paper also covers both cases. Because the TDG scheme has discontinuous elements, we propose to couple it to a surface discretization of the GIBC using continuous finite elements. We prove convergence of the resulting scheme and demonstrate it with two numerical examples.