An interior–exterior Schwarz algorithm and its convergenceUn algorithme de Schwarz intérieur–extérieur et sa convergence

In this work we study the solution of Laplace's equation in a domain with holes by an iteration consisting of splitting the problem in an exterior one, around the holes, plus an interior problem in the unholed domain. We show the existence of a decomposition of the solution when the exterior problem is represented by means of a single-layer protential. Also, for the three-dimensional case and with some adjustments for the two-dimensional case, we prove convergence of the method by writing the iteration as a Jacobi iteration for an operator equation and studying the spectrum of the iteration operator. To cite this article: R. Celorrio et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 923–926.     

On conservative approximation by linear polynomial operators an extension of the Bernstein's operator

In this work we study linear polynomial operators preserving some consecutive i-convexities and leaving invariant the polynomials up to a certain degree. First, we study the existence of an incom patibility between the conservation of certain i-convexities and the invariance of a space of polynomials. Interpolation properties are obtained and a theorem by Berens and DeVore about the Bernstein's operator is extended. Finally, from these results a generalized Bernstein's operator is obtained. This work was supported by Junta de Andalucia. Grupo de investigación: Matemática Aplicada. Código: 1107     

Relaxing the hypotheses of Bielak–MacCamy’s BEM–FEM coupling

In this paper we show that the quasi-symmetric coupling of finite and boundary elements of Bielak and MacCamy can be freed of two very restricting hypotheses that appeared in the original paper: the coupling boundary can be taken polygonal/polyhedral and coupling can be done using the normal stress instead of the pseudostress. We will do this by first considering a model problem associated to the Yukawa equation, where we prove how compactness arguments can be avoided to show stability of Galerkin discretizations of a coupled system in the style of Bielak–MacCamy’s. We also show how discretization properties are robust in the continuation parameter that appears in the formulation. This analysis is carried out using a new and very simplified proof of the ellipticity of the Johnson–Nédélec BEM–FEM coupling operator. Finally, we show how to apply the techniques that we have fully developed in the model problem to the linear elasticity system.     

Design of free-space optical interconnects using two Gabor superlenses

We propose a novel design of micro-optical devices based on multi-aperture compound insect eyes, which transfer a point-to-point multichannel free space signal combined with a diffraction grating. The system is inspired in the refractive superposition compound eyes configuration known as Gabor superlens (GSL) using microlens arrays. A switching function and wave division multiplexing are achieved by introducing a diffraction grating placed in the global focus of the system. The source characteristics, either coherent or incoherent, influence the device performance.     

Two-dimensional optical micro-scanner on silicon technology

A two-dimensional optical micro-scanner, which main components are two mobile flat and a concave micro-mirrors, is designed such that, all optical components can be fabricated on the same substratum. The optical parameters, which physical dimensions are between 50 and 500 μm, are obtained within the geometrical optics. The optical performance is evaluated by means of the MTF and Rayleigh resolution criteria, given 80% of modulation for a frequency of 8 cycles/mm with a Gaussian source, the resolution limit is 30 μm.     

Design of artificial apposition compound eye with cylindrical micro-doublets

We present the design of a non-conventional optical system that uses cylindrical micro-doublets (CMD), integrated in an artificial apposition compound eye configuration (AACE). We show some designs of an ultra-thin objective inspired in fly eyes. These designs can give options to create new technologies that will process information in a different and effective way as usual. This process will be carried out by means of sampling the object with an array of multiple micro lenses using a certain value of the acceptance angle, and processing the optical signal to obtain partial images that will be part of a global one. The objective has the advantage of having a smaller size, a wider field of view, and an acceptable image quality compared with some conventional systems. Design parameters of the AACE and optical performance of the CMD are reported.     

Convergence analysis of a high-order Nyström integral-equation method for surface scattering problems

In this paper we present a convergence analysis for the Nyström method proposed in [J Comput Phys 169 (1):80–110, 2001] for the solution of the combined boundary integral equation formulations of sound-soft acoustic scattering problems in three-dimensional space. This fast and efficient scheme combines FFT techniques and a polar change of variables that cancels out the kernel singularity. We establish the stability of the algorithms in the $L^2$ norm and we derive convergence estimates in both the $L^2$ and $L^\infty $ norms. In particular, our analysis establishes theoretically the previously observed super-algebraic convergence of the method in cases in which the right-hand side is smooth.     

Theoretical aspects of the application of convolution quadrature to scattering of acoustic waves

In this paper we address several theoretical questions related to the numerical approximation of the scattering of acoustic waves in two or three dimensions by penetrable non-homogeneous obstacles using convolution quadrature (CQ) techniques for the time variable and coupled boundary element method/finite element method for the space variable. The applicability of CQ to waves requires polynomial type bounds for operators related to the operator Δ − s ² in the right half complex plane. We propose a new systematic way of dealing with this problem, both at the continuous and semidiscrete-in-space cases. We apply the technique to three different situations: scattering by a group of sound-soft and -hard obstacles, by homogeneous and non-homogeneous obstacles.     

Crouzeix–Raviart boundary elements

This paper establishes a foundation of non-conforming boundary elements. We present a discrete weak formulation of hypersingular integral operator equations that uses Crouzeix–Raviart elements for the approximation. The cases of closed and open polyhedral surfaces are dealt with. We prove that, for shape regular elements, this non-conforming boundary element method converges and that the usual convergence rates of conforming elements are achieved. Key ingredient of the analysis is a discrete Poincaré–Friedrichs inequality in fractional order Sobolev spaces. A numerical experiment confirms the predicted convergence of Crouzeix–Raviart boundary elements. Norbert Heuer is supported by Fondecyt-Chile under grant no. 1080044. F.-J. Sayas is partially supported by MEC-FEDER Project MTM2007-63204 and Gobierno de Aragón (Grupo Consolidado PDIE).