Section: Scientific Foundations

Finite element and finite volume methods

Conservation Laws, Anti-Diffusive Schemes, Viscous Flows, Control, Turbulence, Finite element methods, Finite volume methods

Control in Fluid Mechanics

Flow control techniques are widely used to improve the performances of planes or vehicles, or to drive some internal flows arising for example in combustion chambers. Indeed, they can sensibly reduce energy consumption, noise disturbances, or prevent the flow from undesirable behaviors.

E. Creusé is involved in the development of open and closed active flow control, with applications to recirculation in engines or blood flows.

Numerical Methods for Viscous Flows

Numerical investigations are very useful to check the behavior of systems of equations modelling very complicate dynamics. In order to simulate the motion of mixtures of immiscible fluids having different densities, a recent contribution of the team was to develop an hybrid Finite Element / Finite Volume scheme for the resolution of the variable density 2D incompressible Navier-Stokes equations. The main points of this work were to ensure the consistency of the new method [56] as well as its stability for high density ratios [54] . In order to answer these questions, we have developed a MATLAB code and a C++ code. In the following of this work, C. Calgaro and E. Creusé now have in mind the following objectives, in collaboration with T. Goudon (team COFFEE, Inria Sophia-Antipolis) :

• Distribute the matlab version of the code (with an accurate documentation and a graphic interface) to promote new collaborations in the domain and compare alternative numerical solution methods (for instance to compare updating LU factorizations, see [55] );

• To generalize the stability results obtained in [54] for the scalar transport equation to the full 2D Euler system, in particular very low density values density (near vacuum);

• Complete the C++ code to treat more general hydrodynamic models (combustion theory, transport of pollutants). We plan to check the behavior of the equations (typically the Kazhikhov-Smagulov model of powder-snow avanlanches) in the regime when the current existence theory does not apply, and extend our kinetic asymptotic-based schemes to such problems.

A posteriori error estimators for finite element methods

A posteriori estimates, finite element methods

The team works on a posteriori error estimators for finite element methods, applied to the resolution of several partial differential equations. The objective is to derive useful tools in order to control the global error between the exact solution and the approximated one (reliability of the estimator), and to control the local error leading to adaptive mesh refinement strategies (efficiency of the estimator).

More specifically, E. Creusé works on the derivation of some "reconstruction estimators" based on gradient averaging, for diffusion problems (with S. Nicaise, LAMAV, Valenciennes), the Reissner-Mindlin system (PhD of É. Verhille), and the Maxwell equations (PhD of Z. Tang).