Section: New Results

Homogenization and related topics

Participants : Ronan Costaouec, Claude Le Bris, Frédéric Legoll, William Minvielle, Mathias Rousset, Florian Thomines.

The homogenization of (deterministic) non periodic systems is a well known topic. Although well explored theoretically by many authors, it has been less investigated from the standpoint of numerical approaches (except in the random setting). In collaboration with X. Blanc and P.-L. Lions, C. Le Bris has introduced in [17] a possible theory, giving rise to a numerical approach, for the simulation of multiscale nonperiodic systems. The theoretical considerations are based on earlier works by the same authors (derivation of an algebra of functions appropriate to formalize a theory of homogenization). The numerical endeavour is completely new. Promising results have been obtained on a simple case of a periodic system perturbed by a localized defect. Ongoing works consider other configurations, such as for instance an interface between two different crystalline phases.

A theme closely related to homogenization theory and on which several members of the project team have worked a lot in the past few years is the passage from discrete (atomistic) mechanics to continuum mechanics. In this direction, C. Le Bris, in collaboration with X. Blanc and P.-L. Lions, has established in [18] the rigorous continuum limit of the Newton equations of motion for some simple cases of one-dimensional atomistic system.

The project-team also has pursued its efforts in the field of stochastic homogenization of elliptic equations, aiming at designing numerical approaches that both are pratically relevant and keep the computational workload limited.

An interesting case in that context is when the randomness comes as a small perturbation of the deterministic case. As previously shown by earlier works of the project-team, this situation can indeed be handled with a dedicated approach, which turns out to be far more efficient than the standard approach of stochastic homogenization. A final component of the work completed by Florian Thomines during his PhD thesis has concerned the application of Reduced Basis techniques to that specific context of weakly stochastic homogenization problems. In particular, the approach has been adapted in [39] to efficiently compute the terms of the expansion previously developed by A. Anantharaman and C. Le Bris to approximate a certain category of weakly random homogenization problems. It has been demonstrated that the reduced basis technique is very helpful in this particular context and indeed allows for a speed up of the computation. Another application of the same technique, for a slightly different category of models (still in the framework of weakly random homogenization problems) originally derived by X. Blanc, P.-L. Lions and C. Le Bris, has also been explored. The difficulty, there, is to compute the various corrector equations that parametrically depend on the macroscopic location of the microstructure and the particular realization of that microstructure. The problem is definitely amenable to reduced basis techniques, as demonstrated by some preliminary tests, but definite conclusions on the general validity of the approach are yet to be obtained.

The team has also proceeded to address, from a numerical viewpoint, the case when the randomness is not small. In that case, using the standard homogenization theory, one knows that the homogenized tensor, which is a deterministic matrix, depends on the solution of a stochastic equation, the so-called corrector problem, which is posed on the whole space ${ℝ}^d$. This equation is therefore delicate and expensive to solve. In practice, the space ${ℝ}^d$ is truncated to some bounded domain, on which the corrector problem is numerically solved. In turn, this yields a converging approximation of the homogenized tensor, which happens to be a random matrix. For a given truncation of ${ℝ}^d$, the team has shown in [14] that the variance of this matrix can be reduced using the technique of antithetic variables. F. Legoll and W. Minvielle are currently extending this technique to nonlinear, convex homogenization problems.

In addition, C. Le Bris, F. Legoll, W. Minvielle and M. Rousset are currently investigating the possibility to use other variance reduction approaches, such as control variate techniques. A promising idea is to use the weakly stochastic model previously introduced by A. Anantharaman and C. Le Bris (in which a periodic model is perturbed by a rare stochastic perturbation) to build a control variate model. The preliminary results that have already been obtained are very encouraging.

Another contribution in stochastic homogenization is the following. C. Le Bris, in collaboration with X. Blanc and P.-L. Lions, has recently introduced a variant of the classical random homogenization. For that variant, as often in random homogenization, the homogenized matrix is again defined from a so-called corrector function, which is the solution to a problem set on the entire space. F. Legoll and F. Thomines have described and proved the almost sure convergence of an approximation strategy based on truncated versions of the corrector problem in [64] . F. Legoll and F. Thomines have also established, in the one-dimensional case, a convergence result on the residual process, defined as the difference between the solution to the highly oscillatory problem and the solution to the homogenized problem.

From a numerical perspective, the Multiscale Finite Element Method is a classical strategy to address the situation when the homogenized problem is not known (e.g. in difficult nonlinear cases), or when the scale of the heterogeneities, although small, is not considered to be zero (and hence the homogenized problem cannot be considered as an accurate enough approximation). The extension of this strategy to the stochastic case, when the tensor describing the properties of the material is the sum of a periodic term and a small random term, has been studied by C. Le Bris, F. Legoll and F. Thomines [36] . A method with a much smaller computational cost than the original MsFEM in the stochastic setting has been proposed. Provided the stochastic perturbation is indeed small, the proposed method is as accurate as the original one. The work [36] also provides a complete analysis of the approach, extending that available for the deterministic setting. Such analysis often rely on the rate of convergence of the two scale expansion (in the sense of homogenization theory) of the solution to the highly oscillatory elliptic partial differential equation. Such a result is classic for periodic homogenization. In generic stochastic homogenization, the rate can be arbitrary small, depending on the rate with which the correlations of the random coefficient vanish. C. Le Bris, F. Legoll and F. Thomines have established in [40] such a result for weakly stochastic homogenization, using asymptotic properties of the Green function of the elliptic operator $Lu=-\text{div}\left(A\nabla u\right)$ (where $A$ is a periodic, coercive and bounded matrix), established by F. Legoll in collaboration with X. Blanc and A. Anantharaman [15] .

Still in the framework of the Multiscale Finite Element approach, F. Thomines has further investigated, in collaboration with Y. Efendiev and J. Galvis (Texas A&M University), the use of Reduced Basis methods. They have considered an extension of the MsFEM approach, well suited to the high contrast case, i.e. the case when the ratio between the maximum and the minimum values of the heterogeneous coefficient is large. The main idea of this extension is to complement the standard MsFEM basis functions with the eigenfunctions (associated to the first small eigenvalues) of a local eigenvalue problem. In [30] , Y. Efendiev, J. Galvis and F. Thomines have considered the case when the problem depends on an additional parameter, and have shown how to use the Reduced Basis approach to more efficiently compute the eigenfunctions mentioned above.

Even in simple deterministic cases, there is actually still room for improvement in many different directions for the MsFEM approach. In collaboration with A. Lozinski (University of Toulouse and now at the University of Besançon) who visited the team-project repeatedly during the year, F. Legoll and C. Le Bris have introduced and studied a variant of MsFEM that considers Crouzeix-Raviart type elements on each mesh element. The continuity across edges (or facets) of the (multiscale) finite element basis set functions is enforced only weakly, using fluxes rather than point values. The approach has been analyzed (combining classical arguments from homogenization theory and finite element theory) and tested on simple, but highly convincing cases [35] . In particular, an elliptic problem set on a domain with a huge number of perforations has been considered in [62] . The variant developed outperforms all existing variants of MsFEM. A follow up on this work, in collaboration with U. Hetmaniuk (University of Washington in Seattle, two-week visitor in the project-team in the Spring of 2012), consists in the study of multiscale advection-diffusion problems. Such problems are possibly advection dominated and a stabilization procedure is therefore required. How stabilization interferes with the multiscale character of the equation is an unsolved mathematical question worth considering for numerical purposes.

Still another question related to homogenization theory that is investigated in the group is the following. Consider an elliptic equation, say in divergence form, with a highly oscillatory matrix coefficient. Is it possible to approximate the boundary value problem for different right hand sides using a similar problem with a constant matrix coefficient? How can this “best” constant matrix approximating the oscillatory problem be constructed in an efficient manner? How is this matrix related to the homogenized matrix, in the limit of infinitely rapidly oscillatory coefficients? Current work is directed towards solving such issues.