An analytical verification of a quasicontinuum method

In recent years several engineering models, e.g., in the context of elasticity theory, were justified by a discrete to continuum analysis, i.e., by a passage from discrete/atomistic systems to continuum problems. For variational problems this involves the notion of Γ-convergence. I will talk about some analytical aspects of the so-called quasicontinuum method. This is a computational scheme that has been successfully applied to, e.g., continua with cracks. The idea there is to describe the material with an atomistic model close to the crack tips and with a continuum model away from the crack tips, i.e., away from singularities. This reduces the complexity of atomistic simulations. In joint work with M. Schäffner we analyze the validity of this quasicontinuum approximation. To this end we compare the limiting behavior of the quasicontinuum approximation with the limiting behavior of a fully atomistic system in the context of fracture mechanics by means of Γ-convergence techniques. This yields in particular that the discretization in the continuum region has an impact on the validity of the quasicontinuum approximation.