Scale Convergence in Homogenization

Spectral stochastic two‐scale convergence method for parabolic PDEs

“…The method was applied to transport equation with incompressible velocity field in [12] and [15]. The two-scale convergence method was extended to the case of non periodic oscillations by Mascarenhas and Toader [16]. The concept of stochastic two-scale convergence in the mean has been introduced in Bourgeat et al [17].…”

On general two-scale convergence and its application to the characterization of G-limits

“…See also the careful investigation of the theoretical fundaments and properties of periodic two-scale convergence in [13] by Lukkassen et al Cioranescu et al investigate a new approach, unfolding, closely related to two-scale convergence, in [7], see also [18] by Nechvátal. In [14], Mascarenhas and Toader introduce a concept called "scale-convergence" for Young measures. Further refinements to the method of two-scale convergence are made by Nguetseng in [20].…”

Multi-scale Young measures

“…We will restrict attention here, on this initial work, to typical situations and defer more complicated problems for the future. The idea of analyzing joint or coupled Young measures to derive more information on oscillating sequences has also been specifically studied for various reasons (nearly always related to homogenization issues for differential equations) and in various contexts in [12], [14] and [19]. Yet the full power of the multi-scale Young measure being capable of representing weak limits of compositions of almost any kind and not just of a particular convenient (multiplicative) type has not been explored (see Section 4).…”