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Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the $p$-version

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Hiptmair, R., Moiola, A. and Perugia, I. (2011) Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the $p$-version. SIAM Journal on Numerical Analysis (SINUM), 49 (1). pp. 264-284. ISSN 0036-1429

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To link to this article DOI: 10.1137/090761057

Abstract/Summary

Plane wave discontinuous Galerkin (PWDG) methods are a class of Trefftz-type methods for the spatial discretization of boundary value problems for the Helmholtz operator $-\Delta-\omega^2$, $\omega>0$. They include the so-called ultra weak variational formulation from [O. Cessenat and B. Després, SIAM J. Numer. Anal., 35 (1998), pp. 255–299]. This paper is concerned with the a priori convergence analysis of PWDG in the case of $p$-refinement, that is, the study of the asymptotic behavior of relevant error norms as the number of plane wave directions in the local trial spaces is increased. For convex domains in two space dimensions, we derive convergence rates, employing mesh skeleton-based norms, duality techniques from [P. Monk and D. Wang, Comput. Methods Appl. Mech. Engrg., 175 (1999), pp. 121–136], and plane wave approximation theory.

Item Type:Article
Refereed:Yes
Divisions:Faculty of Science > School of Mathematical and Physical Sciences > Department of Mathematics and Statistics
ID Code:28020
Publisher:Society for Industrial and Applied Mathematics

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