## Section: New Results

### Homogenization

Participants : Xavier Blanc, Virginie Ehrlacher, Olga Gorynina, Rémi Goudey, Claude Le Bris, Frédéric Legoll, Adrien Lesage, Pierre-Loïk Rothé.

In homogenization theory, members of the project-team have pursued their ongoing systematic study of perturbations of periodic problems (by local and nonlocal defects). This has been done in several different directions.

#### Deterministic non-periodic systems

For linear elliptic equations with highly oscillating coefficients, X. Blanc and C. Le Bris have recently developed, in collaboration with P.-L. Lions (Collège de France), a theory in the case of periodic problems with local defects. In particular, the existence of a corrector function for such problems has been shown. More details on the quality of approximation achieved by their theory have been recently provided. The fact that a corrector exists with suitable properties indeed allows one to quantify the rate of convergence of the two-scale expansion (which uses that corrector) to the actual exact solution, as the small homogenization parameter $\epsilon $ vanishes. In that spirit, some of these works by X. Blanc and C. Le Bris, in collaboration with M. Josien (former PhD student in the team, now at MPI Leipzig, Germany), have been presented in [12], [13].

Also in the context of homogenization theory, O. Gorynina, C. Le Bris and F. Legoll have explored the question of how to determine the homogenized coefficient of heterogeneous media without explicitly performing an homogenization approach. This work is a follow-up on earlier works by C. Le Bris and F. Legoll in collaboration with K. Li and next S. Lemaire over the years. During the year, O. Gorynina, C. Le Bris and F. Legoll have mathematically studied a computational approach initially introduced by R. Cottereau (CNRS Marseille). This approach combines, in the Arlequin framework, the original fine-scale description of the medium (modelled by an oscillatory coefficient) with an effective description (modelled by a constant coefficient) and optimizes upon the coefficient of the effective medium to best fit the response of a purely homogeneous medium. In the limit of asymptotically infinitely fine structures, the approach yields the value of the homogenized coefficient. The aim is to mathematically study the problem and to investigate how to improve on the practical algorithm, in order to obtain a procedure as efficient as possible. Results will be presented in a couple of manuscripts in preparation.

#### Stochastic homogenization

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

Using 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 ${\mathbb{R}}^{d}$. This equation is therefore delicate and expensive to solve, and the team has proposed, over the past years, many approaches to improve on the computation of the homogenized tensor.

Besides the averaged behavior of the oscillatory solution ${u}_{\epsilon}$ on large space scales (which is given by its homogenized limit), a question of interest is to describe how ${u}_{\epsilon}$ fluctuates. This question has been investigated in the PhD thesis of P.-L. Rothé, both from a theoretical and a numerical viewpoints. First, theoretical results have been obtained for a weakly stochastic setting (where the coefficient is the sum of a periodic coefficient and a small random perturbation). It has been shown that, at the first order and when $\epsilon $ is small, the localized fluctuations (characterized by a test function $g$) of ${u}_{\epsilon}$ are Gaussian. The corresponding variance depends on the localization function $g$, on the right-hand side $f$ of the problem satisfied by ${u}_{\epsilon}$, and on a fourth order tensor $Q$ which is defined in terms of the corrector. Since the corrector function is challenging to compute, so is $Q$. A numerical approach (based on using the standard truncated corrector problem) has been designed to approximate $Q$ and its convergence has been proven, again in a weakly stochastic setting. All these theoretical results critically depend on detailed properties of the Green function associated to the periodic operator. Second, numerical experiments in more general settings (i.e. full stochastic case) following the same approach have been performed, in order to investigate the generality of the obtained results. First, the convergence of the approximation of $Q$ has been monitored. Second, it has been checked that the localized fluctuations of ${u}_{\epsilon}$ indeed become Gaussian when $\epsilon $ decreases, and that their variance can be related to $Q$. These promising numerical results, which are consistent with the theoretical results obtained in the weakly stochastic setting, are presented in a manuscript in preparation.

#### Multiscale Finite Element approaches

From a numerical perspective, the Multiscale Finite Element Method (MsFEM) 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 a sufficiently accurate approximation).

During the year, several research tracks have been pursued in this general direction.

The MsFEM approach uses a Galerkin approximation on a pre-computed basis, obtained by solving local problems mimicking the problem at hand at the scale of mesh elements, with carefully chosen right-hand sides and boundary conditions. The initially proposed version of MsFEM uses as basis functions the solutions to these local problems, posed on each mesh element, with null right-hand sides and with the coarse P1 elements as Dirichlet boundary conditions. Various improvements have next been proposed, such as the *oversampling* variant, which solves local problems on larger domains and restricts their solutions to the considered element. In collaboration with U. Hetmaniuk (University of Washington in Seattle, USA), C. Le Bris, F. Legoll and P.-L. Rothé have completed the study of a MsFEM method improved differently. They have considered a variant of the classical MsFEM approach with enrichments based on Legendre polynomials, both in the bulk of the mesh elements and on their interfaces. A convergence analysis of this new variant has been performed. In addition, residue type a posteriori error estimators have been proposed and certified, leading to a numerical strategy where the degree of enrichment is *locally* adapted in order to reach, at the smallest computational cost, a given error. The promising numerical results are currently being collected in a manuscript in preparation.

Many numerical analysis studies of the MsFEM are focused on obtaining a priori error bounds. In collaboration with L. Chamoin, who was on leave in the project-team a few years ago from ENS Cachan, members of the project-team have been working on a posteriori error analysis for MsFEM approaches, with the aim of developing error estimation and adaptation tools. They have extended to the MsFEM case an approach that is classical in the computational mechanics community for single scale problems, and which is based on the so-called Constitutive Relation Error (CRE). Once a numerical solution ${u}_{h}$ has been obtained, the approach needs additional computations in order to determine a divergence-free field as close as possible to the exact flux $k\nabla u$. In the context of the MsFEM, it is important to be able to perform all expensive computations in an offline stage, independently of the right-hand side. The standard CRE approach has thus been adapted to that context. In the recent work [47], the approach has also been adapted towards the design of adaptive algorithms for specific quantities of interest (in the so-called “goal-oriented” setting). It provides an accurate estimation of the error, and leads to a discretization which is efficiently tailored to the specific quantity under consideration.

One of the perspectives of the team, through the PhD thesis of A. Lesage, is the development of Multiscale Finite Element Methods for thin heterogeneous plates. The fact that one of the dimension of the domain of interest scales as the typical size of the heterogeneities within the material induces theoretical and practical difficulties that have to be carefully taken into account (see [37]). The first steps of the work of V. Ehrlacher, F. Legoll and A. Lesage, in collaboration with A. Lebée (Ecole des Ponts) have consisted in studying the homogenized limit (and the two-scale expansion) of problems posed on thin heterogeneous plates. After having considered the case of a diffusion equation, the more challenging case of elasticity has been studied. In the so-called membrane case (that is, when the loading is in the in-plane directions), an approximation result for the two-scale expansion has been obtained. Several MsFEM variants have been proposed and compared numerically. The results will be presented in a forthcoming manuscript.