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.
Lingua originaleEnglish
pagine (da-a)1035-1055
Numero di pagine21
RivistaComputational and Applied Mathematics
Volume34
Stato di pubblicazionePublished - 2015

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Helmholtz equation
Constrained optimization
Constrained Optimization Problem
Helmholtz Equation
Unbounded Domain
Infinity
Transparent Boundary Conditions
Radiation Condition
Convergence Estimates
Chebyshev's Method
Functional Integral
Refraction
Collocation Method
Numerical Algorithms
Convergence Properties
Convergence Results
Minimization Problem
Boundary conditions
Radiation
Numerical Examples

All Science Journal Classification (ASJC) codes

  • Computational Mathematics
  • Applied Mathematics

Cita questo

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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.",
keywords = "Helmholtz equation; Minimization of integral functionals; Spectral methods; Transparent boundary conditions; Computational Mathematics; Applied Mathematics",
author = "Vincenzo Sciacca and Francesco Gargano",
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AB - 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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