A Trefftz Discontinuous Galerkin method for time-harmonic waves with a generalized impedance boundary condition

ABSTRACT

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.

COMMUNICATED BY:

• Yongzhi Xu

KEYWORDS:

• Helmholtz equation
• Trefftz
• discontinuous Galerkin
• generalized impedance boundary condition
• error estimate
• artificial boundary condition

AMS SUBJECT CLASSIFICATION:

• 65N30

Acknowledgements

Peter Monk and Shelvean Kapita acknowledge the support of the IMA, University of Minnesota during the special year ‘Mathematics and Optics’.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research of Peter Monk is partially supported by NSF [grant number: DMS-1619904]. The research of Virginia Selgas is partially supported by MCI project number [MTM2013-43671-P].