The Quasicontinuum method [Ortiz, Phillips, Tadmor; 1996] is a  
numerical coarse-graining technique used for the atomistic simulation  
of materials at a microscopic and mesoscopic scale. It uses a finite  
element method to remove unnecessary degrees of freedom from the  
problem.
The talk is divided into 3 parts: First, I will give an introduction  
to the Quasicontinuum method and its applications. Next, I will  
discuss the challenges in its analysis and approximation theory: non- 
uniqueness of solutions and non-convexity of the energy functional. I  
offer a framework in which the QC method can be analyzed, which is  
based on proximal point algorithms (a time-discrete gradient flow)  
and $\lambda$-convexity. As an application, I present a residual- 
based a-posteriori error estimate of a one-dimensional model problem  
and, if time permits, outline its higher dimensional version.