A projection algorithm for fluid–structure interaction problems with strong added-mass effect

This Note aims at introducing a semi-implicit coupling scheme for fluid–structure interaction problems with a strong added-mass effect. Our main idea relies on the splitting of added-mass, viscous effects and geometrical/convective non-linearities, through a Chorin–Temam projection scheme within the fluid. We state some theoretical stability results, in the linear case, and provide some numerical experiments. The main interest of the proposed scheme is its efficiency compared to the implicit approach. To cite this article: M.A. Fernández et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

An added-mass free semi-implicit coupling scheme for fluid–structure interaction

In this Note we propose a semi-implicit coupling scheme for the numerical simulation of fluid–structure interaction systems involving a viscous incompressible fluid. The scheme is stable irrespectively of the so-called added-mass effect and allows for conservative time-stepping within the structure. The efficiency of the scheme is based on the explicit splitting of the viscous effects and geometrical/convective non-linearities, through the use of the Chorin–Temam projection scheme within the fluid. Stability relies on the implicit treatment of the pressure stresses and on the Nitsche's treatment of the viscous coupling. To cite this article: M. Astorino et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Estimations intérieures avec régularité optimale pour un modèle de plaques en élasticité linéaireInterior estimates with optimal regularity for a plate model in linear elasticity

We are interested in the 3d–2d passage for an asymptotically thin plate in linear elasticity. The classical approach by asymptotic expansion gives an error estimate on the displacements in H¹, assuming the volumic forces at least of regularity L² (and more for certain components). In our work we apply the regularity theory for solutions of elliptic equations. This approach gives, for a new model of Kirchhoff–Love of higher order, an error estimate in H² assuming volumic forces only in L², which is optimal. We also get some interior error estimates in W^k,p, C^k,α. To cite this article: R. Monneau, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 207–212.     

Estimateur a posteriori en norme L∞ pour les équations elliptiquesL∞-a posteriori error estimator for elliptic equations

In this Note, we show that modification of Bank–Wieser estimator introduce an L^∞-a posteriori error estimator for conforming and nonconforming methods. We prove, without saturation assumption nor comparison with residual estimators, the equivalence with the L^∞ error. To cite this article: A. Agouzal, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 411–415.     

On Zienkiewicz–Zhu error estimators for Maxwell's equations

We consider a posteriori Zienkiewicz–Zhu (ZZ) type error estimators for the Maxwell equations. The main tool is the use of appropriate recovered values of the electric field and its curl. To cite this article: S. Nicaise, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Reduced-basis approximation of the viscous Burgers equation: rigorous a posteriori error bounds

We present rigorous, sharp, and inexpensive a posteriori error bounds for reduced-basis approximations of the viscosity-parametrized Burgers equation. There are two critical ingredients: the Brezzi, Rappaz and Raviart (Numer. Math. 36 (1980) 1–25) framework for analysis of approximations of nonlinear elliptic partial differential equations; and offline/online computational procedures for efficient calculation of the necessary continuity and stability constants, and of the dual norm of the residual. Numerical results confirm the performance of the error bounds. To cite this article: K. Veroy et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Analysis of curvature influence on effective boundary conditionsAnalyse de l'influence de la courbure sur les conditions aux limites équivalentes

The aim of this work is the construction of effective boundary conditions (wall laws) for elliptic problems defined in domains with curved, rough boundaries with periodic wrinkles. We present error estimates for first and second order approximations, and a numerical test. To cite this article: A. Madureira, F. Valentin, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 499–504.     

Periodic multisequences with large error linear complexity

Generalizing the theory of k-error linear complexity for single sequences over a finite field, Meidl et al. (J. Complexity 23(2), 169–192 (2007)) introduced three possibilities of defining error linear complexity measures for multisequences. A good keystream sequence must possess a large linear complexity and a large k-error linear complexity simultaneously for suitable values of k. In this direction several results on the existence, and lower bounds on the number, of single sequences with large k-error linear complexity were proved in Meidl and Niederreiter (Appl. Algebra Eng. Commun. Comput. 14(4), 273–286 (2003)), Niederreiter (IEEE Trans. Inform. Theory 49(2), 501–505 (2003)) and Niederreiter and Shparlinski (In: Paterson (ed.) 9th IMA International Conference on Cryptography and Coding (2003)). In this paper we discuss analogous results for the case of multisequences. We also present improved bounds on the error linear complexity and on the number of sequences satisfying such bounds for the case of single sequences.     

Choix du paramètre de pénalisation pour la discrétisation par éléments finis des équations de Navier–Stokes

We consider the Navier–Stokes equations, discretized by a penalization method and finite elements. The aim of this Note is to prove a posteriori error estimates which allow for an optimal choice of the penalty parameter, specially for adaptive meshes. To cite this article: C. Bernardi et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Estimation localement suroptimale et adaptative de la densitéLocally superoptimal and adaptive projection density estimators

We study a data-driven version of the density projection estimator in a general framework. We show that this estimator reaches a superoptimal rake on a dense set in the density class, and a quasi-optimal rake elsewhere. This set can be chosen by the statistician, and the superoptimal speed is reached for integrated quadratic error and almost sure uniform convergence. An adaptive version of the estimator is also considered. To cite this article: D. Bosq, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 591–595.     

Multivariate data-driven k-NN function estimation

In Bhattacharya and Mack (Ann. Statist. 15 (1987), 976–994), it was shown (among other things) that adapting for the optimal choice of k in univariate k-nearest neighbor density and regression estimation is feasible using weak convergence techniques. We now show that the same holds true for the multivariate case. Our results parallel Krieger and Pickands (Ann. Statist. 9 (1981), 1066–1078) and Mack and Müller (J. Multivariate Anal. 23 (1987), 169–182) for adaptive multivariate kernel density, respectively, regression, estimation.     

Estimateur hiérarchique robuste pour un problème de perturbation singulière

In this Note, we show that a modified and simplified version of the estimator of Bank–Weiser can be used to define a robust a posteriori error estimator for singularly perturbed problem. We prove without comparison with a residual estimator or saturation assumption, the equivalence of the estimator with the error in the energy norm and the robusteness with respect to the diffusion coefficient. To cite this article: B. Achchab et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Estimating multidimensional density functions for random variables in Wiener space

The Malliavin–Thalmaier (Springer Finance, Springer-Verlag, Berlin, 2006) formula was introduced for the simulation of high dimensional probability density functions. However, when this integration by parts formula is applied directly in computer simulations, we show that it is unstable. We propose an approximation to the Malliavin–Thalmaier formula. In this Note, we find the order of the bias and the variance of the approximation error, and we apply the Malliavin–Thalmaier formula for the calculation of Greeks in finance. To cite this article: A. Kohatsu-Higa, K. Yasuda, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

The entropy condition and the admissibility of shocks

The author proposed (Trans. Amer. Math. Soc.199 (1974), 89–112) the extended entropy condition (E) and solved the Riemann problem for general 2 × 2 conservation laws. The Riemann problem for 3 × 3 gas dynamics equations was treated by the author (J. Differential Equations18 (1975), 218–231). In this paper we justify condition (E) by the viscosity method in the spirit of Gelfand [Uspehi Mat. Nauk14 (1959), 87–158]. We show that a shock satisfies condition (E) if and only if the shock is admissible, that is, it is the limit of progressive wave solutions of the associated viscosity equations. For the “genuinely nonlinear” 2 × 2 conservation laws, Conley and Smoller [Comm. Pure Appl. Math.23 (1970), 867–884] proved that a shock satisfies Lax's shock inequalities [cf. Comm. Pure Appl. Math.14 (1957), 537–566] if and only if it is admissible. In this paper, we consider systems that are not necessarily genuinely nonlinear.     

A non-overlapping domain decomposition method for the Stokes equations via a penalty term on the interfaceDécomposition de domaines sans superposition pour les équations de Stokes via une pénalisation dans l'interface

The purpose of this Note is to perform a theoretical analysis of the domain decomposition method introduced in [2]. We motivate and introduce an improvement of this method and carry out the analysis when it is applied to solving the Stokes equations. Our method is based on a penalty term on the interface between subdomains that enforces the appropriate transmission conditions and may be seen as variation of the Robin method. We obtain strong convergence results for velocity and pressure in the standard H¹ and L² norms and precise rates of convergence, together with error estimates. These error estimates are of optimal order with respect to the precision of the interpolation. We conclude with some numerical tests. To cite this article: T. Chacón Rebollo, E. Chacón Vera, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 221–226.     

Mixed finite element methods for general quadrilateral grids

We study a new mixed finite element of lowest order for general quadrilateral grids which gives optimal order error in the H(div)-norm. This new element is designed so that the H(div)-projection Π_h satisfies ∇ · Π_h = P_hdiv. A rigorous optimal order error estimate is carried out by proving a modified version of the Bramble-Hilbert lemma for vector variables. We show that a local H(div)-projection reproducing certain polynomials suffices to yield an optimal L²-error estimate for the velocity and hence our approach also provides an improved error estimate for original Raviart-Thomas element of lowest order. Numerical experiments are presented to verify our theory.     

Estimation d'erreur et éclatement en homogénéisation périodiqueError estimate and unfolding for periodic homogenization

This Note deals with the error estimate in problems of periodic homogenization. The methods used are those of the periodic unfolding method. The error estimate is obtained without any supplementary hypothesis of regularity on correctors. To cite this article: G. Griso, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 333–336.     

A general reduction scheme for the time-dependent Born–Oppenheimer approximation

We construct a general reduction scheme for the study of the quantum propagator of molecular Schrödinger operators with smooth potentials. This reduction is made up to infinitely (resp. exponentially) small error terms with respect to the inverse square root of the mass of the nuclei, depending on the C^∞ (resp. analytic) smoothness of the interactions. Then we apply this result to the case when an electronic level is isolated from the rest of the spectrum of the electronic Hamiltonian. To cite this article: A. Martinez, V. Sordoni, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 185–188.     

A three field stabilized finite element method for the Stokes equationsUne méthode d'éléments finis stabilisée à trois champs pour les équations de Stokes

We consider in this work the boundary value problem for Stokes equations on a two dimensional domain in cases where non-standard boundary conditions are given. We study the cases where pressure and normal or tangential components of the velocity are given in different parts of the boundary and solve the problem with a minimal regularity. We introduce the problem and its variational formulation which is a mixed one. The principal unknowns are the pressure and the vorticity, the multiplier is the velocity. We present the numerical discretization which needs some stabilization. We prove the convergence and the behavior of the a priori error estimates. Some numerical tests are also presented. To cite this article: M. Amara et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 603–608.     

Domain decomposition for a finite volume method on non-matching grids

We are interested in a robust and accurate domain decomposition method with Robin interface conditions on non-matching grids using a finite volume discretization. We introduce transmission operators on the non-matching grids and define new interface conditions of Robin type. Under a compatibility assumption, we show the equivalence between Robin interface conditions and Dirichlet–Neumann interface conditions and the well-posedness of the global and local problems. Two error estimates are given in terms of the discrete H¹-norm: one in O(h^1/2) with operators based on piecewise constant functions and the other in O(h) (as in the conforming case) with operators using a linear rebuilding. Numerical results are given. To cite this article: L. Saas et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).