Transparent boundary conditions in a Discontinuous Galerkin Trefftz method

TL;DR

This paper introduces a novel transparent boundary condition for a discontinuous Galerkin Trefftz method, reducing reflections in electromagnetic simulations by penalizing incoming wave components, and demonstrates spectral convergence and dissipative behavior.

Contribution

It proposes a new boundary condition using polynomial plane wave bases within a DG Trefftz framework, improving wave reflection minimization in electromagnetic simulations.

Findings

Spectral convergence observed in numerical tests

Reduced parasitic reflections compared to traditional methods

Theoretical explanation of dissipative behavior

Abstract

The modeling and simulation of electromagnetic wave propagation is often accompanied by a restriction to bounded domains which requires the introduction of artificial boundaries. The corresponding boundary conditions should be chosen in order to minimize parasitic reflections. In this paper, we investigate a new type of transparent boundary condition for a discontinuous Galerkin Trefftz finite element method. The choice of a particular basis consisting of polynomial plane waves allows us to split the electromagnetic field into components with a well specified direction of propagation. The reflections at the artificial boundaries are then reduced by penalizing components of the field incoming into the space-time domain of interest. We formally introduce this concept, discuss its realization within the discontinuous Galerkin framework, and demonstrate the performance of the resulting approximations by numerical tests. A comparison with first order absorbing boundary conditions, that are frequently used in practice, is made. For a proper choice of basis functions, we observe spectral convergence in our numerical test and an overall dissipative behavior for which we also give some theoretical explanation.