Nonparametric estimation of the density of the regression noise

This Note presents an estimator of the density of the error in a homoscedastic regression model, based on model selection methods, and propose a bound for the quadratic integrated risk. To cite this article: S. Plancade, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Interior error estimate for periodic homogenization

The aim of this Note is to give interior error estimates for problems in periodic homogenization, by using the periodic unfolding method. The interior error estimates are obtained by transposition without any supplementary hypothesis of regularity on correctors. This error is of order ?. To cite this article: G. Griso, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

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.     

Residual error estimators for the time-dependent Stokes equations

We present a posteriori residual error estimators for the approximate time-dependent Stokes model Chorin–Temam (Chorin, Math. Comp. 23 (1969) 341–353) projection scheme using a conforming finite element discretization. We prove a global upper bound and local lower bounds for the error on the velocity field only. To cite this article: N. Kharrat, Z. Mghazli, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

A posteriori residual error estimation of a cell-centered finite volume method

We present an a posteriori residual error estimator for the Laplace equation using a cell-centered finite volume method in the plane. For that purpose we associate to the approximated solution a kind of Morley interpolant. The error is then the difference between the exact solution and this Morley interpolant. The residual error estimator is based on the jump of normal and tangential derivatives of the Morley interpolant. The equivalence between the discrete H¹-seminorm of the error and the residual error estimator is proved. The proof of the upper error bound uses the Helmholtz decomposition of the broken gradient of the error and some quasi-orthogonality relations. To cite this article: S. Nicaise, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

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).     

A residual based a posteriori estimator for the reaction-diffusion problem

A residual based a posteriori estimator for the reaction-diffusion problem is introduced. We show that the estimator gives both an upper and a lower bound to error. Numerical results are presented. To cite this article: M. Juntunen, R. Stenberg, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Roundoff-error analysis of a new class of conjugate-gradient algorithms

We perform the rounding-error analysis of the conjugate-gradient algorithms for the solution of a large system of linear equations Ax=b where Ais an hermitian and positive definite matrix. We propose a new class of conjugate-gradient algorithms and prove that in the spectral norm the relative error of the computed sequence {x_k} (in floating-point arithmetic) depends at worst on ζк^{$\text{3}\text{2}$}, where ζ is the relative computer precision and к is the condition number of A. We show that the residual vectors r_k=Ax_k-b are at worst of order ζк?vA?v ?vx_k?v. We p oint out that with iterative refinement these algorithms are numerically stable. If ζк ² is at most of order unity, then they are also well behaved.     

Estimation d'erreur pour une loi de conservation scalaire multidimensionelle approchée par un schéma implicite de volumes finis

In this Note we present an error estimate for the approximate solution of the nonlinear hyperbolic equation u_t + div(ƒ(u(x, t))v(x)) = 0 by an implicit finite volume scheme. We show that the error is of order √k +√h, where h and k are respectively the sizes of the space and the time steps parameters. A generalisation of this result to an arbitrary consistent monotone numerical flux is presented in [4]. The convergence of this scheme is possible with k/h (CFL condition) going to infinity.     

Adaptive mesh for algebraic orthogonal subscale stabilization of convective dispersive transport

We derive a residual a posteriori error estimator for the algebraic orthogonal subscales stabilization of convective dispersive transport equation. The estimator yields upper bound on the error which is global and lower bound that is local. Numerical studies show the behaviour of the error indicator and how it is robust to deal with singularities. To cite this article: B. Achchab et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

A posteriori error estimation in the conforming finite element method based on its local conservativity and using local minimization

We present in this Note fully computable a posteriori error estimates allowing for accurate error control in the conforming finite element discretization of pure diffusion problems. The derived estimates are based on the local conservativity of the conforming finite element method on a dual grid associated with simplex vertices rather than directly on the Galerkin orthogonality. To cite this article: M. Vohralík, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

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 monotonic evaluation of lower bounds for inf-sup stability constants in the frame of reduced basis approximations

For accurate a posteriori error analysis of the reduced basis method for coercive and non-coercive problems, a critical ingredient lies in the evaluation of a lower bound for the coercivity or inf-sup constant. In this short Note, we generalize and improve the successive constraint method first presented by Huynh (2007) by providing a monotonic version of this algorithm that leads to both more stable evaluations and fewer offline computations. To cite this article: Y. Chen et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

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.     

Simple regression in view of elliptical models

For the simple linear model Y = θ1 + βx + ? where the error vector follows the elliptically contoured distribution, we consider the unrestricted, restricted, preliminary test and shrinkage estimators for the intercept parameter, θ when it is suspected that the slope parameter β may be β_o. The exact bias and MSE expressions are derived and the mean-square relative efficiency is taken to determine the relative dominance properties of the proposed estimators in comparison. In the continuation, the optimal level of significance of the preliminary test estimator is tabulated and some graphical result are also displayed.     

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.     

A deficient spline function approximation to systems of first-order differential equations: part 2

The problem of stability was discussed in part 1 of this paper (Appl. Math. Modelling 1983, 7, 380). This part looks at the convergence of the spline approximation of deficiency 3 to systems of first-order differential equations. Convergence is shown for m = 4 and 5. In addition, global error bounds of the form: ∥S⁽ⁱ⁾(x) ? y⁽ⁱ⁾(x)∥∞ = 0(h^m+1?i), i = 0(1)m are presented, together with a computational example which illustrates the convergence of the proposed method.     

Error estimates for three-dimensional Stokes problem with non-standard boundary conditions

This work is devoted to the optimal and a posteriori error estimates of the Stokes problem with some non-standard boundary conditions in three dimensions. The variational formulation is decoupled into a system for the velocity and a Poisson equation for the pressure. The velocity is approximated with curl conforming finite elements and the pressure with standard continuous elements. Next, we establish optimal a posteriori estimates.     

Convergence analysis of finite element methods for H(curl; Ω)-elliptic interface problems

In this article we investigate the analysis of a finite element method for solving H(curl; ??)-elliptic interface problems in general three-dimensional polyhedral domains with smooth interfaces. The continuous problems are discretized by means of the first family of lowest order Nédélec H(curl; ??)-conforming finite elements on a family of tetrahedral meshes which resolve the smooth interface in the sense of sufficient approximation in terms of a parameter ?? that quantifies the mismatch between the smooth interface and the triangulation. Optimal error estimates in the H(curl; ??)-norm are obtained for the first time. The analysis is based on a ??-strip argument, a new extension theorem for H ¹(curl; ??)-functions across smooth interfaces, a novel non-standard interface-aware interpolation operator, and a perturbation argument for degrees of freedom for H(curl; ??)-conforming finite elements. Numerical tests are presented to verify the theoretical predictions and confirm the optimal order convergence of the numerical solution.