Analysis Of An Energy-Based Atomistic/Continuum Approximation Of A Vacancy In The 2D Triangular Lattice

We present an a priori error analysis of a practical energy based atomistic/continuum coupling method (A. V. Shapeev, Multiscale Model. Simul., 9(3): 905-932, 2011) in two dimensions, for finite-range pair-potential interactions, in the presence of vacancy defects. We establish first-order consistency and stability of the method, from which we obtain a priori error estimates in the H-1-norm and the energy in terms of the mesh size and the "smoothness" of the atomistic solution in the continuum region. From these error estimates we obtain heuristics for an optimal choice of the atomistic region and the finite element mesh, as well as convergence rates in terms of the number of degrees of freedom. Our analytical predictions are supported by extensive numerical tests.