Analysis of a Spectral-Galerkin Approximation to the Helmholtz Equation in Exterior Domains

Shen, Jie; Wang, Li-Lian

doi:10.1137/060665737

Cited by 60 publications

(53 citation statements)

References 27 publications

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“…The second group of methods abandon the use of piecewise polynomial trial and test functions and replace them by global polynomials or non-polynomial functions. Methods in this group include spectral methods [45], generalized Galerkin/finite element methods [40,8], partition of unity finite element methods [41], and meshless methods [8]. We also note that another very different and intensively studied approach for high frequency wave computation is geometrical optics, which studies asymptotic (nonlinear) approximations of the Helmholtz equation obtained when the frequency (or wave number) tends infinity.…”

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confidence: 99%

Discontinuous Galerkin Methods for the Helmholtz Equation with Large Wave Number

Feng¹,

Wu²

2009

SIAM J. Numer. Anal.

163

227

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Abstract. This paper develops and analyzes some interior penalty discontinuous Galerkin methods using piecewise linear polynomials for the Helmholtz equation with the first order absorbing boundary condition in the two and three dimensions. It is proved that the proposed discontinuous Galerkin methods are stable (hence well-posed) without any mesh constraint. For each fixed wave number k, optimal order (with respect to h) error estimate in the broken H 1 -norm and sub-optimal order estimate in the L 2 -norm are derived without any mesh constraint. The latter estimate improves to optimal order when the mesh size h is restricted to the preasymptotic regime (i.e., k 2 h 1). Numerical experiments are also presented to gauge the theoretical result and to numerically examine the pollution effect (with respect to k) in the error bounds. The novelties of the proposed interior penalty discontinuous Galerkin methods include: first, the methods penalize not only the jumps of the function values across the element edges but also the jumps of the normal and tangential derivatives; second, the penalty parameters are taken as complex numbers of positive imaginary parts so essentially and practically no constraint is imposed on the penalty parameters. Since the Helmholtz problem is a non-Hermitian and indefinite linear problem, as expected, the crucial and the most difficult part of the whole analysis is to establish the stability estimates (i.e., a priori estimates) for the numerical solutions. To the end, the cruxes of our analysis are to establish and to make use of a local version of the Rellich identity (for the Laplacian) and to mimic the stability analysis for the PDE solutions given in [23,24,35].

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mentioning

confidence: 99%

Discontinuous Galerkin Methods for the Helmholtz Equation with Large Wave Number

Feng¹,

Wu²

2009

SIAM J. Numer. Anal.

163

227

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“…The Legendre-Galerkin approximation for (5.15) and the Fourier Legendre-Galerkin approximation for (5.14) have been studied recently in [50]. We now summarize the corresponding results below.…”

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confidence: 86%

“…Remark 5.3. For bounded-obstacle scattering in three dimensions, the results corresponding to Theorems 5.5 and 5.6 were proved in [30] and [50], respectively. Therefore, a result similar to Theorem 5.7 can be established.…”

Section: Periodic Rough Surface Scatteringmentioning

confidence: 99%

“…It is therefore clear, from a practical point of view, that we should place the artificial boundary in such a way that the computational domain Ω 0 is as small as possible. The effect of the artificial boundary location on the convergence rate is manifested through the norms U n H r (Ω0) in (2.5), more precisely the constant C 1 in the above proof (see Remark 4.2 in [50] for a concrete example).…”

Section: David P Nicholls and Jie Shenmentioning

confidence: 99%

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A Rigorous Numerical Analysis of the Transformed Field Expansion Method

Nicholls¹,

Shen²

2009

SIAM J. Numer. Anal.

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Abstract. Boundary perturbation methods, in which the deviation of the problem geometry from a simple one is taken as the small quantity, have received considerable attention in recent years due to an enhanced understanding of their convergence properties. One approach to deriving numerical methods based upon these ideas leads to Bruno and Reitich's generalization [Proc. Roy. Soc. Edinburgh Sect. A, 122 (1992), pp. 317-340] of Rayleigh and Rice's classical algorithm giving the "method of variation of boundaries" which is very fast and accurate within its domain of applicability. Treating problems outside this domain (e.g., boundary perturbations which are large and/or rough) led Nicholls and Reitich [Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), pp. 1411-1433] to design the "transformed field expansions" (TFE) method, and the rigorous numerical analysis of these recursions is the subject of the current work. This analysis is based upon analyticity estimates for the TFE expansions coupled to the convergence of Fourier-Legendre Galerkin methods. This powerful and flexible analysis is extended to a wide range of problems including those governed by Laplace and Helmholtz equations, and the equations of traveling free-surface ideal fluid flow. 1. Introduction. Perturbation techniques have been a crucial tool for scientists and engineers for hundreds of years, many classical and contemporary books, e.g., [25,4], have been devoted to their explication. One very important subclass of such methods are those designed for problems where the perturbation occurs in its geometry; of particular interest for us are partial differential equations posed on complicated domains which are "nearly" simple (e.g., rectangular, circular, and spherical). Such considerations go back at least as far as Stokes [51] in the context of free-surface ideal fluid mechanics (water waves) and Rayleigh [43] regarding the scattering of linear acoustic waves by an irregular obstacle, for example.Quite recently, this class of "boundary perturbation" (BP) methods has been carefully reexamined for use as highly accurate computational methodologies for the simulation of engineering problems. For instance, for the problem of electromagnetic or acoustic scattering by irregular obstacles, Bruno and Reitich [7,8,9,10,11,12] generalized the method of Rayleigh [43] and Rice [46] to arbitrarily high order and demonstrated that this method can deliver highly accurate answers with computational complexities which match the current state-of-the-art. These methods were largely ignored for many years due to an incomplete understanding of their convergence properties. However, these issues were resolved by Bruno and Reitich [6] (and later clarified by one of the authors and Reitich [31]) using the theory of complex

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“…The numerical analysis of this spectral-element method (4.13) to (4.10)-(4.12) can be carried out in a similar fashion as in [22]. Then, the complete error analysis for the transformed field expansion algorithm presented in this paper follows from the general framework established in [14].…”

Section: Then One Can Rewrite (41)-(44) Asmentioning

confidence: 99%

An Efficient and Stable Spectral-Element Method for Acoustic Scattering by an Obstacle

Shen

2013

E Asian J Appl Math

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Analysis of a Spectral-Galerkin Approximation to the Helmholtz Equation in Exterior Domains

Cited by 60 publications

References 27 publications

Discontinuous Galerkin Methods for the Helmholtz Equation with Large Wave Number

Discontinuous Galerkin Methods for the Helmholtz Equation with Large Wave Number

A Rigorous Numerical Analysis of the Transformed Field Expansion Method

An Efficient and Stable Spectral-Element Method for Acoustic Scattering by an Obstacle

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