Optimal error estimate for the optimization-based atomistic-to-continuum method

Let $\bar{u}^{\rm atc}$ be the atomistic-to-continuum approximation, let $\bar{u}$ be the exact solution, let $\mathcal{L}$ be the computational lattice, let $\mathcal{L}_{\rm c}$ be the continuum region, let $h$ denote the finite-element mesh size, and let $D^2\bar{u}:=(D_\rho D_\sigma\bar{u})_{\rho,\sigma\in\mathcal{R}}$. Error estimate. The energy-norm error satisfies

$$\|D \bar{u}^{\rm atc} - D \bar{u}\|_{\ell^2(\mathcal{L})}^2 \lesssim \|D \bar{u}\|_{\ell^2(\mathbb{Z}^d\setminus\mathcal{L})}^2 + \|h D^2 \bar{u}\|_{\ell^2(\mathcal{L}_{\rm c})}^2 =:{\rm err}^2.$$

Here $X\lesssim Y$ means that there exists a constant $c>0$ such that $X\leq cY$. This predicts that the coupling error is dominated by the domain-truncation and continuum-modeling errors; numerical simulations with a next-nearest-neighbor Lennard–Jones model are reported to support the prediction, but no rigorous proof is supplied here.