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

Authors: Ralf Hiptmair, Andrea Moiola and Ilaria Perugia
Journal: Math. Comp. 82 (2013), 247-268
MSC (2010): Primary 65N15, 65N30, 35Q61
Published electronically: July 3, 2012
MathSciNet review: 2983024
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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.

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Additional Information

Ralf Hiptmair
Affiliation: Seminar for Applied Mathematics, ETH Zürich, 8092 Zürich, Switzerland

Andrea Moiola
Affiliation: Seminar for Applied Mathematics, ETH Zürich, 8092 Zürich, Switzerland
Address at time of publication: Department of Mathematics and Statistics, University of Reading,Whiteknights, P.O. Box 220, Reading RG6 6AX, UK

Ilaria Perugia
Affiliation: Dipartimento di Matematica, Università di Pavia, 27100 Pavia, Italy

Keywords: Time-harmonic Maxwell’s equation, discontinuous Galerkin methods, Trefftz methods, $p$–version error analysis, duality estimates, plane waves
Received by editor(s): February 21, 2011
Received by editor(s) in revised form: September 3, 2011
Published electronically: July 3, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.