Stable computation of the functional variation of the Dirichlet–Neumann operator

Fazioli, Carlo; Nicholls, David P.

doi:10.1016/j.jcp.2009.10.021

Cited by 9 publications

(5 citation statements)

References 28 publications

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“…For more complicated geometries, if the domain shape is a small deformation of a separable geometry, then the observations above suggest that a perturbative approach would be fruitful. In fact, this line of enquiry has been followed by several groups with great success [8,9,12,17,[21][22][23][24][25][26][27][28]30]. The current contribution compares most closely with the previous work of the present authors [17] on the most general conditions under which the DNO depends analytically upon boundary perturbations.…”

Section: Introductionsupporting

confidence: 63%

The domain of analyticity of Dirichlet–Neumann operators

Hu¹,

Nicholls²

2010

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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Dirichlet-Neumann operators arise in many applications in the sciences, and this has inspired a number of studies on their analytical properties. In this paper we further investigate the analyticity properties of Dirichlet-Neumann operators as functions of the boundary shape. In particular, we study the size of the disc of convergence of their Taylor-series representation. For this we use a complexification technique which requires a novel reformulation of the problem, coupled with methods for systems of elliptic partial differential equations. Numerical results to illustrate our theoretical conclusions are presented.

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Section: Introductionsupporting

confidence: 63%

The domain of analyticity of Dirichlet–Neumann operators

Hu¹,

Nicholls²

2010

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…the strong convergence of the series (24)). We accomplish all of this using the framework built by the author in his collaborations with Reitich [29,32,12], Hu [33,34], Taber [35], and Fazioli [36].…”

Section: Analyticitymentioning

confidence: 99%

A rapid boundary perturbation algorithm for scattering by families of rough surfaces

Nicholls¹

2009

Journal of Computational Physics

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a b s t r a c tIn this paper we describe a novel algorithm for the computation of scattering returns by families of rough surfaces. This algorithm makes explicit use of the fact that some scattering profiles of engineering interest (e.g., traveling ocean waves) come in branches parameterized analytically by a bifurcation quantity. Our approach delivers recursions which not only can be implemented to yield a rapid, robust and high-order numerical scheme, but also give a new proof of analyticity of scattering quantities with respect to the bifurcation parameter of the surface family. The real advantage of this new approach is that it computes, in one step, the scattered field for all possible members of the family of surfaces. By contrast, other state-of-the-art schemes must restart when the returns from a new surface are desired, so that the cost of our new approach is greatly advantaged when the number of samples of the family reaches even modest values. Numerical results which verify the accuracy of our approach and demonstrate their utility in computing grating efficiencies scattered by traveling surface ocean waves are presented.

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“…For the computation of G(η) and G η (η), we use the algorithms developed by Nicholls and Fazioli (33;34). The essence of these algorithms is to evaluate the terms G n and G…”

Section: Methodsmentioning

confidence: 99%