Section: New Results

Homogenization

We have three types of results regarding the homogenization theory and its applications.

The first series of results is related to nonlinear elasticity. In [2] , A. Gloria has proved the convergence of a discrete model for rubber towards a nonlinear elasticity theory in collaboration with R. Alicandro and M. Cicalese. The numerical simulation of the model has been addressed within the ARC Disco by A. Gloria, P. La Tallec and M. Vidrascu (project team MACS). Comparisons with mechanical experiments are promising, and related inverse problems have been addressed in the post-doc of M. de Buhan. Two publications are in preparation. Related theoretical results in homogenizaton of nonlinear elasticity models have been obtained by A. Gloria and S. Neukamm (MPI Leipzig) in [15] .

A second type of results concerns a quantitative theory of stochastic homogenization of discrete linear elliptic equations. A breakthrough has been obtained by A. Gloria and F. Otto (MPI Leipzig) in [16] and [17] , who gave the first optimal variance estimate of the energy density of the corrector field for stochastic discrete elliptic equations. The proof makes extensive use of a spectral gap estimate and of deep elliptic regularity theory, bringing in fact the probabilistic arguments to a minimum. This analysis has enabled A. Gloria to propose efficient numerical homogenization methods, both in the discrete and continuous settings [13] , [12] . In [14] , A. Gloria and J. C. Mourrat has pushed the approach forward and introduced new approximation formulas for the homogenized coefficient. In [26] they have considered a more probabilistic approach and given a complete error analysis of a Monte-Carlo approximation of the homogenized coefficients in the discrete case. Work in progress concerns the generalization of the results on discrete elliptic equations to the continuous case.

The third direction of research concerns the periodic homogenization of a coupled elliptic/parabolic system arising in the modeling of nuclear waste storage. This work is in collaboration with the French agency ANDRA. A. Gloria, T. Goudon, and S. Krell have made a complete theoretical analysis of the problem, derived effective equations, and devised an efficient method to solve the effective problem numerically, based on the reduced basis approach. A publication is in preparation. This subject will be continued with the arrival of Z. Habibi.