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A spectral approach to a constrained optimization problem for the Helmholtz equation in unbounded domains

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Abstract

We study some convergence issues for a recent approach to the problem of transparent boundary conditions for the Helmholtz equation in unbounded domains (Ciraolo et al. in J Comput Phys 246:78–95, 2013) where the index of refraction is not required to be constant at infinity. The approach is based on the minimization of an integral functional, which arises from an integral formulation of the radiation condition at infinity. In this paper, we implement a Fourier–Chebyshev collocation method to study some convergence properties of the numerical algorithm; in particular, we give numerical evidence of some convergence estimates available in the literature (Ciraolo in Helmholtz equation in unbounded domains: some convergence results for a constrained optimization problem, 2013) and study numerically the minimization problem at low and mid-high frequencies. Numerical examples in some relevant cases are also shown.

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Notes

  1. We implemented (6) by using a finite element scheme and we noticed that the grid points needed in this case were \(10^2\) times more than in the spectral collocation method (\(10^6\) and \(10^4\) grid points, respectively).

  2. We notice that, thanks to the collocation method, the accuracy of a Fourier–Chebyshev-type scheme is equivalent to the one of a Fourier–Jacobi method when one uses (roughly) the same order of polynomials (Livermore et al. 2007; Mohseni and Colonius 2000; Verkley 1997a, b; Matsushima and Marcus 1995; Boyd 2001).

    Collocation methods for ODEs and PDEs can also be implemented by using different basis, like Hermite or Legendre polynomials (Öztürk and Gülsu 2013; Khader and Sweilam 2013); however, those approaches require a suitable decay at infinity which is not satisfied in our case (see also Shen and Wang 2009 for a more exhaustive reading on the recent advances on the spectral methods).

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Acknowledgments

The authors wish to thank the anonymous referee for his valuable comments and suggestions to improve the quality of the paper. The work of F. Gargano and V. Sciacca has been supported by National Group of Mathematical Physics (GNFM-INDAM). G. Ciraolo has been supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Palermo, Via Archirafi 34, 90123, Palermo, Italy

    Giulio Ciraolo, Francesco Gargano & Vincenzo Sciacca

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  1. Giulio Ciraolo
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  2. Francesco Gargano
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  3. Vincenzo Sciacca
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Correspondence to Giulio Ciraolo.

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Communicated by José Mario Martínez.

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Ciraolo, G., Gargano, F. & Sciacca, V. A spectral approach to a constrained optimization problem for the Helmholtz equation in unbounded domains. Comp. Appl. Math. 34, 1035–1055 (2015). https://doi.org/10.1007/s40314-014-0164-5

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  • DOI: https://doi.org/10.1007/s40314-014-0164-5

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