Application of an Ultra Weak Variational Formulation of Elliptic PDEs to the Two-Dimensional Helmholtz Problem

Cessenat, Olivier; Després, Bruno

doi:10.1137/s0036142995285873

Cited by 334 publications

(409 citation statements)

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“…These local solutions are then patched together to form an approximate global solution. Possible techniques include the partition of unity finite element method [16,17], a Lagrange multiplier technique [20], least squares methods [18,19] or the Ultra Weak Variational Formulation (UWVF) [5][6][7]. It is the last of these techniques that will be the focus of this paper.…”

Section: Introductionmentioning

confidence: 99%

“…The UWVF, which we shall describe precisely in Section 2, particularly equation (2.17), is a variational formulation of the Helmholtz equation due to Cessenat and Després [5,6]. It is based on a mesh of the domain where the Helmholtz equation is to be solved and computes the trace of the solution of the Helmholtz equation and its normal derivative on the skeleton of the mesh (i.e.…”

Section: Introductionmentioning

confidence: 99%

“…Cessenat and Després [5,6] prove an error estimate for the method showing that the solution of the UWVF converges to an appropriate impedance trace of the true solution on the boundary of the domain and in particular they prove that the boundary error is bounded by a suitable best approximation error in the entire domain. They then derive explicit error estimates by analyzing the convergence of the best approximation error.…”

Section: Introductionmentioning

confidence: 99%

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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: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Buffa¹,

Monk²

2008

ESAIM: M2AN

106

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“…which is generally preferred due to the improved conditioning of the problem [6], [13]. The latter form is particularly useful when large-scale problems are solved using iterative methods.…”

Section: Ultra-weak Variational Formulationmentioning

confidence: 99%

“…The enriched methods include, for example, the partition of unity finite element method (PUFEM) [18,3], the generalized finite element method (GFEM) [21,2], which is a combination of the classical finite element method and the partition of unity method, and discontinuous enrichment method (DEM) [9]. A purely plane wave basis has been used, for example, with the ultra-weak variational formulation [6,7], discontinuous Galerkin method [10] and the least-squares method [19]. The aim of this study is to compare the PUFEM and UWVF for 2D Helmholtz problems.…”

Section: Introductionmentioning

confidence: 99%

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 Partition of Unity Method

Babuška

Melenk

1997

Int. J. Numer. Meth. Engng.

2,031

762

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A new finite element method is presented that features the ability to include in the finite element space knowledge about the partial differential equation being solved. This new method can therefore be more efficient than the usual finite element methods. An additional feature of the partition-of-unity method is that finite element spaces of any desired regularity can be constructed very easily. This paper includes a convergence proof of this method and illustrates its efficiency by an application to the Helmholtz equation for high wave numbers. The basic estimates for a posteriori error estimation for this new method are also proved.KEY WORDS: finite element method; meshless finite element method; finite element methods for highly oscillatory solutions

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Application of an Ultra Weak Variational Formulation of Elliptic PDEs to the Two-Dimensional Helmholtz Problem

Cited by 334 publications

References 4 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

Comparison of two wave element methods for the Helmholtz problem

The Partition of Unity Method

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