Computational Aspects of the Ultra-Weak Variational Formulation

Huttunen, Tomi; Monk, Peter; Kaipio, Jari P.

doi:10.1006/jcph.2002.7148

Cited by 125 publications

(169 citation statements)

References 14 publications

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“…This agrees with usual statement of the UWVF in terms of unknown functions on F h , see [7], Formula 19, [10], Formula (1.4), and [22], Formula 10. Matching (2.8), (2.9), and (2.10), (2.11), we see that the original UWVF by Cessenat and Després [10] is recovered by choosing α = 1/2, β = 1/2, γ = 0, δ= 1/2.…”

Section: Equation (26) Simply Becomesmentioning

confidence: 44%

Plane wave discontinuous Galerkin methods: Analysis of theh-version

2009

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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.Mathematics Subject Classification. 65N15, 65N30, 35J05.

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Section: Equation (26) Simply Becomesmentioning

confidence: 44%

Plane wave discontinuous Galerkin methods: Analysis of theh-version

2009

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“…We have found it necessary to choose p j depending on the element K j and wave number k in order to control ill-conditioning in certain matrices in the formulation [13].…”

Section: Derivation Of the Uwvfmentioning

confidence: 46%

“…We shall give more details of their results later in this paper. Extensive numerical experiments by Cessenat and Després, as well as by others [12][13][14], shows that the method converges throughout the domain of computation. The main purpose of this paper is to prove global convergence of the UWVF applied to the Helmholtz equation in the case where the medium is not absorbing as is often the case for scattering calculations.…”

Section: Introductionmentioning

confidence: 45%

Error estimates for the Ultra Weak Variational Formulation of the Helmholtz equation

Buffa¹,

Monk²

2008

ESAIM: M2AN

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106

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Abstract. The Ultra Weak Variational Formulation (UWVF) of the Helmholtz equation provides avariational framework suitable for discretization using plane wave solutions of an appropriate adjoint equation. Currently convergence of the method is only proved on the boundary of the domain. However substantial computational evidence exists showing that the method also converges throughout the domain of the Helmholtz equation. In this paper we exploit the fact that the UWVF is essentially an upwind discontinuous Galerkin method to prove convergence of the solution in the special case where there is no absorbing medium present. We also provide some other estimates in the case when absorption is present, and give some simple numerical results to test the estimates. We expect that similar techniques can be used to prove error estimates for the UWVF applied to Maxwell's equations and elasticity.Mathematics Subject Classification. 65N15, 65N30, 35J05.

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“…3, we compare these two approaches for the duct problem at the wave number κ = 40. The UWVF results are computed by solving the discrete equation in the form (13). Results for both the propagating (m = 12) and the evanescent mode (m = 13) show that the most memory efficient increase in accuracy is obtained via the p-refinement.…”

Section: Wave Propagation In a Duct With Rigid Wallssupporting

confidence: 39%

“…The reason for this is, however, most likely in the two step inversion of the UWVF equation used in (13). It is shown earlier (see e.g.…”

Section: Wave Propagation In a Duct With Rigid Wallsmentioning

confidence: 44%

Comparison of two wave element methods for the Helmholtz problem

Huttunen

Gamallo

Astley

2008

Commun. Numer. Meth. Engng.

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In comparison with low‐order finite element methods (FEMs), the use of oscillatory basis functions has been shown to reduce the computational complexity associated with the numerical approximation of Helmholtz problems at high wave numbers. We compare two different wave element methods for the 2D Helmholtz problems. The methods chosen for this study are the partition of unity FEM (PUFEM) and the ultra‐weak variational formulation (UWVF). In both methods, the local approximation of wave field is computed using a set of plane waves for constructing the basis functions. However, the methods are based on different variational formulations; the PUFEM basis also includes a polynomial component, whereas the UWVF basis consists purely of plane waves. As model problems we investigate propagating and evanescent wave modes in a duct with rigid walls and singular eigenmodes in an L‐shaped domain. Results show a good performance of both methods for the modes in the duct, but only a satisfactory accuracy was obtained in the case of the singular field. On the other hand, both the methods can suffer from the ill‐conditioning of the resulting matrix system. Copyright © 2008 John Wiley & Sons, Ltd.

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Computational Aspects of the Ultra-Weak Variational Formulation

Cited by 125 publications

References 14 publications

Plane wave discontinuous Galerkin methods: Analysis of theh-version

Plane wave discontinuous Galerkin methods: Analysis of theh-version

Error estimates for the Ultra Weak Variational Formulation of the Helmholtz equation

Comparison of two wave element methods for the Helmholtz problem

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