Abstract. We present a comprehensive error analysis of two prototypical atomistic-tocontinuum coupling methods of blending type: the energy-based and the force-based quasicontinuum methods.Our results are valid in two and three dimensions, for finite range many-body interactions (e.g., EAM type), and in the presence of lattice defects (we consider point defects and dislocations). The two key ingredients in the analysis are (i) new force and energy consistency error estimates; and (ii) a new technique for proving energy norm stability of a/c couplings that requires only the assumption that the exact atomistic solution is a stable equilibrium.

We formulate an atomistic-to-continuum coupling method based on blending atomistic and continuum forces. Our precise choice of blending mechanism is informed by theoretical predictions. We present a range of numerical experiments studying the accuracy of the scheme, focusing in particular on its stability. These experiments confirm and extend the theoretical predictions, and demonstrate a superior accuracy of B-QCF over energy-based blending schemes. IntroductionAtomistic-to-continuum coupling methods (a/c methods) have been proposed to increase the computational efficiency of atomistic computations involving the interaction between local crystal defects with long-range elastic fields [6,7,15,18,22,29,30,40]; see [26] for a recent review of a/c coupling methods and their numerical analysis. Energy-based methods in this class, such as the quasicontinuum model (denoted QCE [41]), exhibit spurious interfacial forces ("ghost forces") even under uniform strain [8,39]. The effect of the ghost force on the error in computing the deformation and the lattice stability by the QCE approximation has been analyzed in [8][9][10]31], where lattice stability refers to the positive definiteness of the Hessian matrix of the total potential energy. The development of more accurate energy-based a/c methods is an ongoing process [5,15,20,34,37,38,40].An alternative approach to a/c coupling is the force-based quasicontinuum (QCF) approximation [7,11,12,25,29], but the non-conservative and indefinite equilibrium equations make the iterative solution and the determination of lattice stability more challenging [12][13][14]. Indeed, it is an open problem whether the (sharp-interface) QCF method is stable in dimension greater than one. Although some recent results in this direction exist [24], it is still unclear to what extent they can be extended for general atomistic domains and in the presence of defects.Many blended a/c coupling methods have been proposed in the literature, e.g., [1-4, 16, 23, 35, 36, 42]. In [21], we formulated a blended force-based quasicontinuum (B-QCF) method, similar to the method proposed in [25], which smoothly blends the forces of the atomistic and continuum Xingjie Li, 182 George St., Providence, RI 02912, USA, xingjie li@brown.edu M. Luskin (Corresponding Author),