Report 2009-20

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

R. Hiptmair, A. Moiola and I. Perugia

Abstract: Plane wave discontinuous Galerkin methods (PWDG) 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.~Despr\'es, Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz equation, 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, A least squares method for the Helmholtz equation, Computer Methods in Applied Mechanics and Engineering, 175 (1999), pp.~121--136], and plane wave approximation theory.

Keywords: Helmholtz equation, wave propagation, discontinuous Galerkin (DG) methods, plane waves, p--version error analysis, duality estimates

Paper: Available as PDF (553 KB) or as hardcopy to order reports@sam.math.ethz.ch.