INTERNSHIP FOR A MASTER STUDENT

HOMOGENIZATION OF PARABOLIC EQUATIONS WITH
LARGE DRIFT

Research area : partial differential equations, numerical analysis, homogenization.

Place : CMAP (Centre de Math´ematiques Appliqu´ees), Ecole Polytechnique, 91128, Palaiseau Cedex.

Adviser : Grégoire Allaire (gregoire.allaire@polytechnique.fr).

Duration of the internship : 4 to 6 months. This internship can be followed by a PhD thesis.

Topic :

Homogenization is the theory of averaging partial differential equations in heterogeneous domains. The goal of this internship is to study the homogenization of parabolic equations in the special case where a large drift or velocity appears in the limit or homogenized problem. This phenomenon has many important applications like convection-diffusion in porous media, neutron diffusion in nuclear reactors, or bio-motors. At the beginning of the internship the student will perform the following tasks:

  1. make some numerical computations to find explicit coeffcients of a parabolic system leading to a large drift,

  2. make a theoretical comparison between the homogenization results obtained in the case of Dirichlet boundary conditions [1], [2] and those in the case of Neumann boundary conditions [3].

A good acquaintance with the analysis of partial differential equations is necessary. A basic knowledge of numerical analysis is sufficient. No specific knowledge of homogenization is required.

References

[1] G. Allaire, A.-L. Raphael, Homogenization of a convection-diffusion model with reaction in a porous medium, C. R. Acad. Sci. Paris, Série I, 344, pp.523-528 (2007).

[2] Y. Capdeboscq, Homogenization of a diffusion equation with drift, C.R.Acad.Sci.Paris, t.327, Srie I, p.807-812, 1998.

[3] B. PERTHAME, P. E. SOUGANIDIS Asymmetric potentials and motor effect: a homogenization approach, preprint R08046, Laboratoire J.-L. Lions, Université Paris 6 (2008).