Multiscale methods coupling atomistic and continuum mechanics: some examples of mathematical analysis

Many numerical methods coupling a discrete description of matter with a continuum one have been recently proposed in the mechanics literature. This contribution aims at introducing this field in a mathematical perspective, at an elementary level. We first review some modelling questions. We next carry on a mathematical analysis on a toy example of an atomistic to continuum coupling method. This example presents the same coupling features as some more advanced methods. On the other hand, its simplicity makes its analysis easier. Specific difficulties linked with the coupling of models at different scales are highlighted. We conclude these notes by reviewing some general questions frequently debated in the field, pointing out the main directions that have been followed at the front of research, and discussing some open problems