Claude Le Bris: How to deal with nonperiodicity in multiscale problems? Toward a theory of defects.

We will present some recent mathematical and numerical contributions related to nonperiodic multiscale problems. A typical, simple setting is that of a scalar elliptic equation with an highly oscillatory coefficient. The difficulty is that this coefficient is not assumed periodic. It models a structure with a set of embedded localized defects, or a structure that, although not periodic, enjoys nice geometrical features. The purpose is then to construct theoretical settings, and next numerical approaches providing an efficient and accurate approximation of the solution in several possible cases of practical interest. The questions raised ranged from the theory of elliptic PDEs, Calderon-Zygmund operators, homogenization theory, to more numerical issues related to finite element methods.