This paper summarizes the homogenization process of rough, hydrodynamic lubrication problems governed by the Reynolds equation used to describe compressible liquid flow. Here, the homogenized equation describes the limiting result when the wavelength of a modeled surface roughness goes to zero. The lubricant film thickness is modeled by one part describing the geometry/shape of the bearing and a periodic part describing the surface topography/roughness. By varying the periodic part as well as its wavelength, we can try to systematically investigate the applicability of homogenization on this type of problem. The load carrying capacity is the target parameter; deterministic solutions are compared to homogenized by this measure. We show that the load carrying capacity rapidly converges to the homogenized results as the wavelength decreases, proving that the homogenized solution gives a very accurate representation of the problem when real surface topographies are considered

This thesis is devoted to homogenization theory and some generalizations of Rothe's method to non-cylindrical domains. It consists of two introductory chapters and four papers. Chapters 1 and 2 serve as a self-contained overview to the theory of homogenization. In the introduction we present the idea behind homogenization theory and in the second chapter some further results in homogenization theory are presented and some of the homogenization results from chapter 1 are proved. Paper A deals with a numerical study of stochastic homogenization and paper B deals with some generalizations of Rothe's method to non-cylindrical domains. Paper C is devoted to numerical analysis of the convergence in homogenization of composites and finally the degeneracy in stochastic homogenization is considered in the paper D.

This note makes the link between theoretical results on stochastic homogenization and effective computation of averaged coefficients for diffusion operators in random media. Examples of how to construct relevant random media and numerical results on the effective coefficients are given.