Numerical Homogenization: Survey, New Results, and Perspectives

Abstract. These notes give a state of the art of numerical homogenization methods for linear elliptic equations. The guideline of these notes is analysis. Most of the numerical homogenization methods can be seen as (more or less different) discretizations of the same family of continuous approximate problems, which H-converges to the homogenized problem. Likewise numerical correctors may also be interpreted as approximations of Tartar’s correctors. Hence the convergence analysis of these methods relies on the H-convergence theory. When one is interested in convergence rates, the story is different. In particular one first needs to make additional structure assumptions on the heterogeneities (say periodicity for instance). In that case, a crucial tool is the spectral interpretation of the corrector equation by Papanicolaou and Varadhan. Spectral analysis does not only allow to obtain convergence rates, but also to devise efficient new approximation methods. For both qualitative and quantitative properties, the development and the analysis of numerical homogenization methods rely on seminal concepts of the homogenization theory. These notes contain some new results.