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. I will present results with M. Schäffner in a one-dimensional
setting with interaction potentials that are of Lennard-Jones type and
thus have a convex-concave shape. This allows the formation of cracks.
The focus will be on finite range interactions. In particular I will
show a method that allows to trace the proof back to earlier proofs in
the case of nearest and next-to-nearest neighbor interactions. We will
discuss fully atomistic systems as well as systems which mimic some
computational quasicontinuum method. This provides a rigorous analytical
understanding of the quasicontinuum method and yields a condition on the
choice of the finite element mesh that ensures a reasonable approximation.http://cvgmt.sns.it/seminar/479/
