Robust equilibrated a posteriori error estimators for the Reissner-Mindlin system

We consider a conforming finite element approximation of the Reissner-Mindlin system. We propose a new robust a posteriori error estimator based on H(div) conforming finite elements and equilibrated fluxes. It is shown that this estimator gives rise to an upper bound where the constant is one up to higher order terms. Lower bounds can also be established with constants depending on the shape regularity of the mesh. The reliability and efficiency of the proposed estimator are confirmed by some numerical tests

A posteriori error estimation for the Stokes problem: Anisotropic and isotropic discretizations

The paper presents a posteriori error estimators for the stationary Stokes problem. We consider anisotropic finite element discretizations (i.e. elements with very large aspect ratio) where conventional, isotropic error estimators fail. Our analysis covers two- and three-dimensional domains, conforming and nonconforming discretizations as well as different elements. This large variety of settings requires different approaches and results in different estimators. Furthermore many examples of finite element pairs that are covered by the analysis are presented. Lower and upper error bounds form the main result with minimal assumptions on the elements. The lower error bound is uniform with respect to the mesh anisotropy with the exception of nonconforming 3D discretizations made of pentahedra or hexahedra. The upper error bound depends on a proper alignment of the anisotropy of the mesh which is a common feature of anisotropic error estimation. In the special case of isotropic meshes, the results simplify, and upper and lower error bounds hold unconditionally. Some of the corresponding results seem to be novel (in particular for 3D domains), and cover element pairs of practical importance. The numerical experiments confirm the theoretical predictions and show the usefulness of the anisotropic error estimators

A posteriori error estimator based on gradient recovery by averaging for discontinuous Galerkin methods

International audienceWe consider some (anisotropic and piecewise constant) diffusion problems in domains of R^2, approximated by a discontinuous Galerkin method with polynomials of any fixed degree. We propose an a posteriori error estimator based on gradient recovery by averaging. It is shown that this estimator gives rise to an upper bound where the constant is one up to some additional terms that guarantee reliability. The lower bound is also established. Moreover these additional terms are negligible when the recovered gradient is super convergent. The reliability and efficiency of the proposed estimator in confirmed by some numerical tests

A posteriori error estimations of a coupled mixed and standard Galerkin method for second order operators

International audienceIn this paper, we consider a discretization method proposed by Wieners and Wohlmuth [26] (see also [16]) for second order operators, which is a coupling between a mixed method in a sub-domain and a standard Galerkin method in the remaining part of the domain. We perform an a posteriori error analysis of residual type of this method, by combining some arguments from a posteriori error analysis of Galerkin methods and mixed methods. The reliability and efficiency of the estimator are proved. Some numerical tests are presented and confirm the theoretical error bounds

Robust residual a posteriori error estimators for the Reissner-Mindlin eigenvalues system

International audienceWe consider a conforming finite element approximation of the Reissner-Mindlin eigenvalue system, for which a robust a posteriori error estimator for the eigenvector and the eigenvalue errors is proposed. For that purpose, we first perform a robust a priori error analysis without strong regularity assumption. Upper and lower bounds are then obtained up to higher order terms that are super convergent, provided that the eigenvalue is simple. The convergence rate of the proposed estimator is confirmed by a numerical test

A posteriori error estimator based on gradient recovery by averaging for convection-diffusion-reaction problems approximated by discontinuous Galerkin methods

International audienceWe consider some (anisotropic and piecewise constant) convection-diffusion-reaction problems in domains of R2, approximated by a discontinuous Galerkin method with polynomials of any degree. We propose two a posteriori error estimators based on gradient recovery by averaging. It is shown that these estimators give rise to an upper bound where the constant is explicitly known up to some additional terms that guarantees reliability. The lower bound is also established, one being robust when the convection term (or the reaction term) becomes dominant. Moreover, the estimator is asymptotically exact when the recovered gradient is superconvergent. The reliability and efficiency of the proposed estimators are confirmed by some numerical tests

Discrete compactness for a discontinuous Galerkin approximation of Maxwell's system

International audienceIn this paper we prove the discrete compactness property for a discontinuous Galerkin approximation of Maxwell's system on quite general tetrahedral meshes. As a consequence, a discrete Friedrichs inequality is obtained and the convergence of the discrete eigenvalues to the continuous ones is deduced using the theory of collectively compact operators. Some numerical experiments confirm the theoretical predictions

Identification of concentrated structures in slightly compressible flows

National audienceIn this paper, different vortex diagnostic methods are compared to obtain a better understanding of boundary layer influence on the transport of vortical structures involving a complete analysis of vorticity, the Vorticity Threshold Criterion (VTC), and the Weiss Criterion (WC). These three techniques are basically confronted to find a suitable understanding of all flow characteristics for a range of laminar to transitional Reynolds numbers. The computations on this dihedral plane are done using a 2D DNS method. The Weiss criterion, coming from the analysis of the incompressible Euler equations is validated and applied to low speed compressible flows (Mach number=0.2)

En recherchant la vague

Œuvres et Recherches 2020:01En recherchant la vague est une installation qui s'inspire de l'autobiographie de Henri Charrière. Ce prisonnier du bagne de Cayenne, en Guyane française, utilisait des techniques élémentaires de comptage afin de se familiariser avec le mouvement des vagues, pour ensuite organiser son évasion. L'installation comprend des objets et des images qui présentent la façon dont des équations mathématiques modélisent dans le temps et dans l'espace le mouvement et l'évolution de fluides