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Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations

by R. Hiptmair and A. Moiola and I. Perugia

(Report number 2011-09)

Abstract
In this paper, we extend to the time-harmonic Maxwell equations the p-version analysis technique developed in [R. Hiptmair, A. Moiola and I. Perugia, Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version, SIAM J. Numer. Anal., 49 (2011), 264-284] for Trefftz-discontinuous Galerkin approximations of the Helmholtz problem. While error estimates in a mesh-skeleton norm are derived parallel to the Helmholtz case, the derivation of estimates in a mesh-independent norm requires new twists in the duality argument. The particular case where the local Trefftz approximation spaces are built of vector-valued plane wave functions is considered, and convergence rates are derived.

Keywords: Time-harmonic Maxwell's equation, discontinuous Galerkin methods, Trefftz methods, p-version error analysis, duality estimates, plane waves

@Techreport{HMP11_62,
  author = {R. Hiptmair and A. Moiola and I. Perugia},
  title = {Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2011-09},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2011/2011-09.pdf },
  year = {2011}
}

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