# Plane wave discontinuous Galerkin methods: Analysis of the h-version

Claude J. Gittelson; Ralf Hiptmair; Ilaria Perugia

ESAIM: Mathematical Modelling and Numerical Analysis (2009)

- Volume: 43, Issue: 2, page 297-331
- ISSN: 0764-583X

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topGittelson, Claude J., Hiptmair, Ralf, and Perugia, Ilaria. "Plane wave discontinuous Galerkin methods: Analysis of the h-version." ESAIM: Mathematical Modelling and Numerical Analysis 43.2 (2009): 297-331. <http://eudml.org/doc/250619>.

@article{Gittelson2009,

abstract = {
We are concerned with a finite element approximation for time-harmonic wave
propagation governed by the Helmholtz equation. The usually oscillatory behavior of
solutions, along with numerical dispersion, render standard finite element methods
grossly inefficient already in medium-frequency regimes. As an alternative, methods
that incorporate information about the solution in the form of plane waves have
been proposed. We focus on a class of Trefftz-type discontinuous Galerkin methods that
employs trial and test spaces spanned by local plane waves. In this paper we give
a priori convergence estimates for the h-version of these plane wave
discontinuous Galerkin methods in two dimensions. To that end, we develop
new inverse and approximation estimates for plane waves
and use these in the context of duality techniques. Asymptotic optimality of the
method in a mesh dependent norm can be established. However, the estimates require
a minimal resolution of the mesh beyond what it takes to resolve the wavelength. We
give numerical evidence that this requirement cannot be dispensed with. It reflects
the presence of numerical dispersion.
},

author = {Gittelson, Claude J., Hiptmair, Ralf, Perugia, Ilaria},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Wave propagation; finite element methods; discontinuous Galerkin methods; plane
waves; ultra weak variational formulation; duality estimates; numerical dispersion.; wave propagation; plane waves; numerical dispersion; Helmholtz equation; Trefftz type local trial spaces; convergence; numerical results},

language = {eng},

month = {2},

number = {2},

pages = {297-331},

publisher = {EDP Sciences},

title = {Plane wave discontinuous Galerkin methods: Analysis of the h-version},

url = {http://eudml.org/doc/250619},

volume = {43},

year = {2009},

}

TY - JOUR

AU - Gittelson, Claude J.

AU - Hiptmair, Ralf

AU - Perugia, Ilaria

TI - Plane wave discontinuous Galerkin methods: Analysis of the h-version

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2009/2//

PB - EDP Sciences

VL - 43

IS - 2

SP - 297

EP - 331

AB -
We are concerned with a finite element approximation for time-harmonic wave
propagation governed by the Helmholtz equation. The usually oscillatory behavior of
solutions, along with numerical dispersion, render standard finite element methods
grossly inefficient already in medium-frequency regimes. As an alternative, methods
that incorporate information about the solution in the form of plane waves have
been proposed. We focus on a class of Trefftz-type discontinuous Galerkin methods that
employs trial and test spaces spanned by local plane waves. In this paper we give
a priori convergence estimates for the h-version of these plane wave
discontinuous Galerkin methods in two dimensions. To that end, we develop
new inverse and approximation estimates for plane waves
and use these in the context of duality techniques. Asymptotic optimality of the
method in a mesh dependent norm can be established. However, the estimates require
a minimal resolution of the mesh beyond what it takes to resolve the wavelength. We
give numerical evidence that this requirement cannot be dispensed with. It reflects
the presence of numerical dispersion.

LA - eng

KW - Wave propagation; finite element methods; discontinuous Galerkin methods; plane
waves; ultra weak variational formulation; duality estimates; numerical dispersion.; wave propagation; plane waves; numerical dispersion; Helmholtz equation; Trefftz type local trial spaces; convergence; numerical results

UR - http://eudml.org/doc/250619

ER -

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