Dans cette thèse on étudie quelques modèles macroscopiques pour des milieux conducteurs ou élastiques fortement hétérogènes et anisotropes obtenus par homogénéisation. Nous considérons le cas de l’homogénéisation périodique. En particulier pour le système de l’élasticité linéarisée modélisant les petites déformations d’un matériau fibré, nous étudions l’effet de l’anisotropie du matériau sur le modèle macroscopique et nous montrons que sous l’effet conjugué des conditions aux limites et de l’anisotropie des fibres, le système modélisant les déplacements à l’échelle macroscopique fait intervenir des termes non standard. Nous considérons plusieurs scalings et deux situations géométriques : dans la première le rayon des fibres cylindriques est du même ordre de grandeur que la taille de la période du milieu et dans la seconde la rayon est petit devant la période. Les résultats obtenus dans les deux cas, indépendants d’hypothèses de symétrie sur le matériau, permettent de retrouver les résultats déjà connus dans le cas de matériaux isotropes. / In this thesis we study some macroscopic models for drivers or elastic media highly heterogeneous and anisotropic obtained by homogenization. We consider the case of periodic homogenization. In particular the system of linearized elasticity modeling small deformations of a fiber material, we study the effect of material anisotropy on the macroscopic model and show that the combined effect of the boundary conditions and the anisotropy of the fiber system modeling movement at the macroscopic scale involves non-standard terms. We consider several scalings and two geometric situations: in the first radius of cylindrical fibers is of the same order of magnitude as the size of the middle period and in the second the radius is small compared to the period. The results obtained in both cases, independent of symmetry assumptions on the material used to find the results already known in the case of isotropic materials.

In this dissertation, we develop multiscale finite element methods and uncertainty quantification technique for Richards' equation, a mathematical model to describe fluid flow in unsaturated porous media. Both coarse-level and fine-level numerical computation techniques are presented. To develop an accurate coarse-scale numerical method, we need to construct an effective multiscale map that is able to capture the multiscale features of the large-scale solution without resolving the small scale details. With a careful choice of the coarse spaces for multiscale finite element methods, we can significantly reduce errors. We introduce several methods to construct coarse spaces for multiscale finite element methods. A coarse space based on local spectral problems is also presented. The construction of coarse spaces begins with an initial choice of multiscale basis functions supported in coarse regions. These basis functions are complemented using weighted local spectral eigenfunctions. These newly constructed basis functions can capture the small scale features of the solution within a coarse-grid block and give us an accurate coarse-scale solution. However, it is expensive to compute the local basis functions for each parameter value for a nonlinear equation. To overcome this difficulty, local reduced basis method is discussed, which provides smaller dimension spaces with which to compute the basis functions. Robust solution techniques for Richards' equation at a fine scale are discussed. We construct iterative solvers for Richards' equation, whose number of iterations is independent of the contrast. We employ two-level domain decomposition pre-conditioners to solve linear systems arising in approximation of problems with high contrast. We show that, by using the local spectral coarse space for the preconditioners, the number of iterations for these solvers is independent of the physical properties of the media. Several numerical experiments are given to support the theoretical results. Last, we present numerical methods for uncertainty quantification applications for Richards' equation. Numerical methods combined with stochastic solution techniques are proposed to sample conductivities of porous media given in integrated data. Our proposed algorithm is based on upscaling techniques and the Markov chain Monte Carlo method. Sampling results are presented to prove the efficiency and accuracy of our algorithm.