An analytical framework for numerical homogenization. Part II: Windowing and oversampling

In a recent paper [Multiscale Model. Simul., 5 (2006), pp. 996-1043], the author has introduced an analytical framework to study the convergence properties of some numerical homogenization methods for elliptic problems. In the applications however, these methods are coupled with windowing or oversampling techniques. In the present work, the author addresses this issue within the latter framework and proves the convergence of the methods with windowing, for convex and quasiconvex energies, in the context of general heterogeneities. This analysis provides us with an interesting variational interpretation of the Petrov-Galerkin formulation of the nonconforming multiscale finite element method for periodic problems.