The Ultra-Weak Variational Formulation for Elastic Wave Problems

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

doi:10.1137/s1064827503422233

Cited by 67 publications

(92 citation statements)

References 19 publications

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“…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: 70%

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

Buffa¹,

Monk²

2008

ESAIM: M2AN

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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Section: Introductionmentioning

confidence: 70%

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

Buffa¹,

Monk²

2008

ESAIM: M2AN

106

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“…This time integration method results, however, in most test cases in a decrease of the discrete energy. Although not addressed in this article, the methodology is expected to apply to other cases, such as the generalized linear system of Huttunen et al [7] and the three-dimensional acoustic equations [2]. We plan to explore these applications in our future research.…”

Section: Resultsmentioning

confidence: 99%

Discontinuous Hamiltonian Finite Element Method for Linear Hyperbolic Systems

Vegt

Bokhove

2008

J Sci Comput

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We develop a Hamiltonian discontinuous finite element discretization of a generalized Hamiltonian system for linear hyperbolic systems, which include the rotating shallow water equations, the acoustic and Maxwell equations. These equations have a Hamiltonian structure with a bilinear Poisson bracket, and as a consequence the phase-space structure, "mass" and energy are preserved. We discretize the bilinear Poisson bracket in each element with discontinuous elements and introduce numerical fluxes via integration by parts while preserving the skew-symmetry of the bracket. This automatically results in a mass and energy conservative discretization. When combined with a symplectic time integration method, energy is approximately conserved and shows no drift. For comparison, the discontinuous Galerkin method for this problem is also used. A variety numerical examples is shown to illustrate the accuracy and capability of the new method.

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“…It was observed that care must be taken of the form in which the discrete UWVF is solved. While the preconditioned form (13) has a lower condition number its accuracy starts to deteriorate before that of the unpreconditioned equation (12).…”

Section: Discussionmentioning

confidence: 99%

“…It is interesting that despite the UWVF matrix system (13) having a condition number two orders lower than the unpreconditioned form (13), the accuracy of the solution of (13) deteriorates earlier than for (12). The reason for this is, however, most likely in the two step inversion of the UWVF equation used in (13).…”

Section: Wave Propagation In a Duct With Rigid Wallsmentioning

confidence: 99%

“…The original work on the UWVF focused mainly on the solution of the 3D Maxwell equations [5]. Subsequently, the method has been also extended for elastic wave problems [12] and fluid-structure interaction problems [11]. On the other hand, the PUFEM has been applied to flow acoustics problems and proved to be far more efficient than standard finite element methods when applied to short wave acoustic propagation on uniform and nonuniform potential flows [2,1].…”

Section: Introductionmentioning

confidence: 99%

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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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The Ultra-Weak Variational Formulation for Elastic Wave Problems

Cited by 67 publications

References 19 publications

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

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

Discontinuous Hamiltonian Finite Element Method for Linear Hyperbolic Systems

Comparison of two wave element methods for the Helmholtz problem

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