Iterative Methods for the Force-based Quasicontinuum Approximation

Force-based atomistic-continuum hybrid methods are the only known pointwise consistent methods for coupling a general atomistic model to a finite element continuum model. For this reason, and due to their algorithmic simplicity, force-based coupling methods have become a popular approach for atomistic-continuum hybrid methods as well as other types of multiphysics model coupling. However, the recently discovered unusual stability properties of the linearized force-based quasicontinuum approximation, especially its indefiniteness, present a challenge to the development of efficient and reliable iterative methods. Using a combination of rigorous analysis and computational experiments, we present a systematic study of the stability and rate of convergence of a variety of linear stationary iterative methods and generalized minimal residual methods (GMRES) for the solution of the linearized force-based quasicontinuum equations.