Speaker: Mitchell Luskin
Title: Mathematical Results and Challenges for the Quasicontinuum Approximation.
Affiliation: University of Minnesota.
Date: Friday, Febuary 23, 2007.
Place and Time: Room 101 - Love Building, 3:35-4:30 pm.
Refreshments: Room 204 - Love Building, 3:00 pm.

Abstract. The local lattice structure for minimum energy configurations of atomistic systems subject to external forces is usually slowly varying except near defects such as dislocations. Quasicontinuum methods efficiently approximate these multiscale features by maintaining atomistic degrees of freedom near defects and coarse-graining the atomistic degrees of freedom in regions where the local lattice structure is nearly uniform through the introduction of representative atoms. The efficiency of quasicontinuum methods has allowed the simulation of more complex problems than can be computed using a completely atomistic model.

We derive and compare several quasicontinuum approximations to a one-dimensional system of atoms that interact by a classical atomistic potential. We prove that the equilibrium equations have a unique solution under suitable restrictions on the loads, and we give a convergence rate for an iterative method to solve the equilibrium equations. Joint work with Matthew Dobson.