A Generalized Finite Element Method for solving the Helmholtz equation in two dimensions with minimal pollution

Babuška, Ivo; Ihlenburg, Frank; Paik, Ellen T.; Sauter, Stéfan A.

doi:10.1016/0045-7825(95)00890-x

Cited by 348 publications

(295 citation statements)

References 7 publications

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“…However, it is known [33] that, for κh=constant, the errors of the finite element solutions deteriorate rapidly as the wavenumber κ increases. This nonrobust behaviour with respect to κ is known as the pollution effect [36][37][38][39][40][41][42]. The pollution effect may be reduced by increasing the order of the elements in the finite element method or using a small enough mesh for the resolution.…”

Section: Numerical Simulationsmentioning

confidence: 99%

Chacterization of a 20 kHz sonoreactor. Part I: analysis of mechanical effects by classical and numerical methods

Sáez

Frías-Ferrer

Iniesta

et al. 2005

Ultrasonics Sonochemistry

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Section: Numerical Simulationsmentioning

confidence: 99%

Chacterization of a 20 kHz sonoreactor. Part I: analysis of mechanical effects by classical and numerical methods

Sáez

Frías-Ferrer

Iniesta

et al. 2005

Ultrasonics Sonochemistry

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“…7 demonstrates that for the specific choice δ PML = 10 −6 , so that kδ PML ∈ I opt (and a single element in the PML region, N PML = 1) the L 2 error behaves like E ∼ N −(p+1) (for p = 1, 2 and 3) as N is increased, indicating that the error is completely controlled by the discretisation of the bulk mesh. We refer to [32,33,34] for a discussion of the optimal convergence rate of numerical solutions obtained from finite element discretisations of the Helmholtz equation and note that the observed convergence rate indicates that the bulk discretisation was always sufficiently fine to avoid dispersion errors. …”

Section: Test Case 1: a One-dimensional Waveguidementioning

confidence: 99%

A parameter-free perfectly matched layer formulation for the finite-element-based solution of the Helmholtz equation

Cimpeanu

Martinsson

Heil

2015

Journal of Computational Physics

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This paper presents a parameter-free perfectly matched layer (PML) method for the finite-element-based solution of the Helmholtz equation. We employ one of Bermúdez et al.'s unbounded absorbing functions for the complex coordinate mapping underlying the PML. With this choice, the only free parameter that controls the accuracy of the numerical solution for a fixed numerical cost (characterised by the number of elements in the bulk and the PML regions) is the thickness of the perfectly matched layer, δ PML . We show that, for the case of planar waves, the absorbing function performs best for PMLs whose thickness is much smaller than the wavelength. We then perform extensive numerical experiments to explore its performance for non-planar waves, considering domain shapes with smooth and polygonal boundaries, different solution types (smooth and singular), and a wide range of wavenumbers, k, to identify an optimal range for the normalised PML thickness, kδ PML , such that, within this range, the error introduced by the presence of the PML is consistently small and insensitive to change. This implies that if the PML thickness is chosen from within this range * Corresponding author URL: http://www3.imperial.ac.uk/people/radu.cimpeanu11 (Radu Cimpeanu), http://www.maths.manchester.ac.uk/~mheil/ (Matthias Heil) Preprint submitted to Journal of Computational PhysicsMay 4, 2015 no further PML optimisation is required, i.e. the method is parameter-free. We characterise the dependence of the error on the discretisation parameters and establish the conditions under which the convergence of the solution under mesh refinement is controlled exclusively by the discretisation of the bulk mesh.

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“…Although significant progress in the understanding of the behavior of numerical methods for Helmholtz problems has been made in the past, a general, full analysis that is explicit in the wave number k and discretization parameters is still not available. Partial results such as sharp estimates for the inf-sup constant of the continuous equations, lower estimates for the convergence rates, one-dimensional analysis by using the discrete Green's function as well as a dispersion analysis for finite element discretizations and generalizations thereof have been derived by many researchers in the past decades (see, e.g., [2,4,6,7,9,10,11,15,17,18,19,22,23,24,25,26,27,28,33,38,43,44]). …”

Section: Introductionmentioning

confidence: 99%

Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions

2010

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Abstract. A rigorous convergence theory for Galerkin methods for a model Helmholtz problem in R d , d ∈ {1, 2, 3} is presented. General conditions on the approximation properties of the approximation space are stated that ensure quasi-optimality of the method. As an application of the general theory, a full error analysis of the classical hp-version of the finite element method (hp-FEM) is presented for the model problem where the dependence on the mesh width h, the approximation order p, and the wave number k is given explicitly. In particular, it is shown that quasi-optimality is obtained under the conditions that kh/p is sufficiently small and the polynomial degree p is at least O(log k).

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A Generalized Finite Element Method for solving the Helmholtz equation in two dimensions with minimal pollution

Cited by 348 publications

References 7 publications

Chacterization of a 20 kHz sonoreactor. Part I: analysis of mechanical effects by classical and numerical methods

Chacterization of a 20 kHz sonoreactor. Part I: analysis of mechanical effects by classical and numerical methods

A parameter-free perfectly matched layer formulation for the finite-element-based solution of the Helmholtz equation

Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions

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