Stability estimates for a class of Helmholtz problems

Hetmaniuk, Ulrich

doi:10.4310/cms.2007.v5.n3.a8

Cited by 91 publications

(118 citation statements)

References 9 publications

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“…Note that a similar result has been established in [33] assuming the computational domain being a polygonal-shaped domain. Similar estimates with different approaches can be found for example in [35] and references therein.…”

supporting

confidence: 60%

A local wave tracking strategy for efficiently solving mid- and high-frequency Helmholtz problems

Amara

Chaudhry

Diaz

et al. 2014

Computer Methods in Applied Mechanics and Engineering

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We propose a procedure for selecting basis function orientation to improve the efficiency of solution methodologies that employ local plane-wave approximations. The proposed adaptive approach consists of a local wave tracking strategy. Each plane-wave basis set within considered elements of the mesh partition is individually or collectively rotated to best align one function of the set with the local propagation direction of the field. Systematic determination of the direction of the field inside the computational domain is formulated as a minimization problem. As the resultant system is nonlinear with respect to the directions of propagation, the Newton method is employed with exact characterization of the Jacobian and Hessian. To illustrate the salient features and evaluate the performance of the proposed wave tracking approach, we present error estimates as well as numerical results obtained by incorporating the procedure into a prototypical plane-wave based approach, the least-squares method (LSM) developed by Monk et al. [1]. The numerical results obtained for the case of a two-dimensional rigid scattering problem indicate that (a) convergence was achievable to a prescribed level of accuracy, even upon initial application of the tracking wave strategy outside the pre-asymptotic convergence region, and (b) the proposed approach reduced the size of the resulting system by up to two orders of magnitude, depending * Corresponding author. Address: Department of Mathematics, California State University, Northridge, 18111 Nordhoff St., Northridge, CA 91330-8131. Tel: +1-818-677-5867 E-mail address: rabia.djellouli@csun.edu Preprint submitted to Comput. Methods Appl. Mech. Engrg. March 7, 2014 on the frequency range, with respect to the size of the standard LSM system.

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supporting

confidence: 60%

A local wave tracking strategy for efficiently solving mid- and high-frequency Helmholtz problems

Amara

Chaudhry

Diaz

et al. 2014

Computer Methods in Applied Mechanics and Engineering

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“…Hence, the assumption (2.7) puts implicit conditions on the domain Ω and the configuration of the boundary components. Sufficient geometric conditions in two or three dimensions that ensure (2.7) with n = 0 can be found in the original work of Melenk [Mel95] (which based on the choice of a particular test function previously used in [MIB96]) and its generalizations [Het02,CF06,Het07,EM12]. Among the known admissible setups are the case of a Robin boundary condition (Γ R = ∂Ω) on a Lipschitz domain Ω [EM12].…”

Section: Model Helmholtz Problemmentioning

confidence: 99%

“…Among the known admissible setups are the case of a Robin boundary condition (Γ R = ∂Ω) on a Lipschitz domain Ω [EM12]. Another example is the scattering of acoustic waves at a sound-soft scatterer occupying the star-shaped polygonal or polyhedral domain Ω D where the Sommerfeld radiation condition is approximated by the Robin boundary condition on the boundary of some artificial convex polygonal or polyhedral domain Ω R ⊃Ω D ; see [Het07]. Given some linear functional g on V , the adjoint problem of (2.3) seeks z ∈ V such that, for any v ∈ V ,…”

Section: Model Helmholtz Problemmentioning

confidence: 99%

Eliminating the pollution effect in Helmholtz problems by local subscale correction

Peterseim

2016

Math. Comp.

115

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Abstract. We introduce a new Petrov-Galerkin multiscale method for the numerical approximation of the Helmholtz equation with large wave number κ in bounded domains in R d . The discrete trial and test spaces are generated from standard mesh-based finite elements by local subscale correction in the spirit of numerical homogenization. The precomputation of the correction involves the solution of coercive cell problems on localized subdomains of size ℓH; H being the mesh size and ℓ being the oversampling parameter. If the mesh size and the oversampling parameter are such that Hκ and log(κ)/ℓ fall below some generic constants and if the cell problems are solved sufficiently accurate on some finer scale of discretization, then the method is stable and its error is proportional to H; pollution effects are eliminated in this regime.

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“…For example, Rellich introduced (1.34) with v = u in order to obtain an expression for the eigenvalues of the Laplacian as an integral over ∂Ω (instead of the usual expression as an integral over Ω used in, e.g., the Rayleigh-Ritz method), and these identities have been used to further study eigenvalues of equations involving the Laplacian in, e.g., [64], [66], [42], [3], [4]. Rellich-type identities have been wellused by the harmonic analysis community (see, e.g., [47 [24], [43], [19], [45]); some of this recent work is discussed in Remarks 3.6 and 4.7 below. (The recent review [17, section 5.3] explains why Rellich-type identities can be used to do these things.…”

Section: The Idea Behind the New Formulationmentioning

confidence: 99%