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This paper outlines the theoretical background of a new approach towards an accurate and well-conditioned perturbative calculation of Dirichlet{Neumann operators (DNOs) on domains that are perturbations of simple geometries. Previous work on the analyticity of DNOs has produced formulae that, as we have found, are very ill-conditioned. We show how a simple… (More)

We present a new algorithm for the numerical solution of problems of electromagnetic or acoustic scattering by large, convex obstacles. This algorithm combines the use of an ansatz for the unknown density in a boundary-integral formulation of the scattering problem with an extension of the ideas of the method of stationary phase. We include numerical… (More)

We present new, stabilized shape-perturbation methods for calculations of scattering from rough surfaces. For practical purposes, we present new algorithms for both low- (first- and second-) and high-order implementations. The new schemes are designed with guidance from our previous results that uncovered the basic mechanism behind the instabilities that… (More)

The analytic dependence of Dirichlet-Neumann operators (DNO) with respect to variations of their domain of definition has been successfully used to devise diverse computational strategies for their estimation. These strategies have historically proven very competitive when dealing with small deviations from exactly solvable geometries, as in this case the… (More)

The Euler equations of free-surface ocean dynamics constitute a model of central importance in fluid mechanics due to the wide range of physical phenomena they are intended to represent, from shoaling and breaking of waves in nearshore regions to energy and momentum transport in the open ocean. From a mathematical perspective, these equations present rather… (More)

In this paper we present results on the stability of perturbation methods for the evaluation of Dirichlet–Neumann operators (DNO) defined on domains that are viewed as complex deformations of a basic, simple geometry. In such cases, geometric perturbation methods, based on variations of the spatial domains of definition , have long been recognized to… (More)

We analyze the conditioning properties of classical shape-perturbation methods for the prediction of scattering returns from rough surfaces. A central observation relates to the identification of significant cancellations that are present in the recurrence relations satisfied by successive terms in a perturbation series. We show that these cancellations are… (More)

In this paper we establish the existence and analyticity of periodic solutions of a classical free-boundary model of the evolution of three-dimensional, capillary–gravity waves on the surface of an ideal fluid. The result is achieved through the application of bifurcation theory to a boundary perturbation formulation of the problem, and it yields… (More)

- Avner Friedman, Fernando Reitich
- 2007

We consider a one-phase quasi-steady Stefan free boundary problem with surface tension, when the initial position of the free boundary is close to the unit sphere in R (2), and expressed in the form r = 1 + 0 (!). It is proved that the problem has a unique global solution with free boundary which is analytic in and which converges exponentially fast, as t !… (More)

This paper presents a high-order accelerated algorithm for the solution of the integral-equation formulation of volumetric scattering problems. The scheme is particularly well suited to the analysis of "thin" structures as they arise in certain applications (e.g., material coatings); in addition, it is also designed to be used in conjunction with existing… (More)