Emmanuel Creusé

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 non-conforming 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.

An hybrid finite volume-finite element method for variable density incompressible flows

This paper is devoted to the numerical simulation of variable density incompressible flows, modeled by the Navier-Stokes system. We introduce an hybrid scheme which combines a finite volume approach for treating the mass conservation equation and a finite element method to deal with the momentum equation and the divergence free constraint. The breakthrough relies on the definition of a suitable footbridge between the two methods, through the design of compatibility condition. In turn, the method is very flexible and allows to deal with unstructured meshes. Several numerical tests are performed to show the scheme capabilities. In particular, the viscous Rayleigh-Taylor instability evolution is carefully investigated.

L∞-stability of vertex-based MUSCL finite volume schemes on unstructured grids: Simulation of incompressible flows with high density ratios

This work is devoted to the design of multi-dimensional finite volume schemes for solving transport equations on unstructured grids. In the framework of MUSCL vertex-based methods we construct numerical fluxes such that the local maximum property is guaranteed under an explicit Courant-Friedrichs-Levy condition. The method can be naturally completed by adaptive local mesh refinements and it turns out that the mesh generation is less constrained than when using the competitive cell-centered methods. We illustrate the effectiveness of the scheme by simulating variable density incompressible viscous flows. Numerical simulations underline the theoretical predictions and succeed in the computation of high density ratio phenomena such as a water bubble falling in air.

Residual-based a posteriori estimators for the A/phi magnetodynamic harmonic formulation of the Maxwell system

This paper is devoted to the derivation of an a posteriori residual-based error estimator for the A/phi magnetodynamic harmonic formulation of the Maxwell system. The weak continuous and discrete formulations are established, and the well-posedness of both of them is addressed. Some useful analytical tools are derived. Among them, an ad-hoc Helmholtz decomposition is proven, which allows to pertinently split the error. Consequently, an a posteriori error estimator is obtained, which is proven to be reliable and locally efficient. Finally, numerical tests confirm the theoretical results.

Residual and equilibrated error estimators for magnetostatic problems solved by finite element method

In finite element computations, the choice of the mesh is crucial to obtain accurate solutions. In order to evaluate the quality of the mesh, a posteriori error estimators can be used. In this paper, we develop residual-based error estimators for magnetostatic problems with both classical formulations in term of potentials used, as well as the equilibrated error estimator. We compare their behaviors on some numerical applications, to understand the interest of each of them in the remeshing process.

Active procedures to control the flow past the Ahmed body with a 25° rear window

Ahmed body with a 25 degree rear window is used to represent a simpli ed car geometry. Two and three-dimensional simulations are performed to analyse the flow behaviour around such a vehicle. Sucking and blowing jets or slots are added on the body to control the flow. The results presented show that good drag reductions are achieved for a good choice of the active procedure.

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

We 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 superconvergent. The reliability and efficiency of the proposed estimator is confirmed by some numerical tests.

Space-Time Residual-Based A Posteriori Estimators for the A−ϕ Magnetodynamic Formulation of the Maxwell System

Abstract In this paper, an a posteriori residual error estimator is proposed for the A−ϕ magnetodynamic Maxwell system given in its potential and space/time formulation and solved by a finite element method. The reliability as well as the efficiency of the estimator are established for several norms. Then, numerical tests are performed, allowing to illustrate the obtained theoretical results.

A posteriori error estimator for harmonic A‐φ formulation

Purpose – In this paper, the aim is to propose a residual‐based error estimator to evaluate the numerical error induced by the computation of the electromagnetic systems using a finite element method in the case of the harmonic A‐φ formulation.Design/methodology/approach – The residual based error estimator used in this paper verifies the mathematical property of global and local error estimation (reliability and efficiency).Findings – This estimator used is based on the evaluation of quantities weakly verified in the case of harmonic A‐φ formulation.Originality/value – In this paper, it is shown that the proposed estimator, based on the mathematical developments, is hardness in the case of the typical applications.

Simulation of low Reynolds number flow control over a backward-facing step using pulsed inlet velocities

Flow control is a rapidly developing area of fluid dynamics that offers the possibility of significant improvements in flow behavior applied to better engineering capabilities. It might be used to postpone the transition from laminar to turbulent regime, to reduce the skin friction drag and to delay the separation. In case studies where there are regions of separation, flow control can be used to encourage earlier reattachment or, more globally, to improve the shedding and transport phenomena in both internal and external flows (see,e.g. ,[ 3]). One of the classical benchmark case studies of the internal separatedflows with a rich comparing possibility to the literature is the flow over a backward-facing step. This flow has been analyzed either for laminar evolutions (see, e.g. ,[ 1, 4, 7]) or for higher Reynolds numbers (see, e.g. ,[ 6]) experimentally and numerically. An advantage of this case study is its quite simplified geometry that can be used to investigate several engineering problems, like flow in the combustion chambers, power plants, biomedical flows, and so forth. In this geometry, the behavior of the recirculation zone is a quantitative measure that should give strong data on the flow. Furthermore, it gives the opportunity to explore various possibilities to manipulate eddies shedding and transport. The flow studied in this work is laminar. The laminar flow is essentially steady and two-dimensional. This property has already been investigated by several authors. Armaly, Durst, Pereira, and Schonung [1] found a spatial variation of about 1 %f or the