Higher-order Far-field Boundary Conditions for Crystalline Defects

Crystalline materials exhibit long-range elastic fields due to the presence of defects, leading to significant domain size effects in atomistic simulations. A rigorous far-field expansion of these long-range fields identifies low-rank structure in the form of a sum of discrete multipole terms and continuum predictors [J. Braun, T. Hudson, and C. Ortner, Arch. Ration. Mech. Anal., 245 (2022), pp. 1437-1490]. We propose a novel numerical scheme that exploits this low-rank structure to accelerate material defect simulations by minimizing the domain size effects. Our approach iteratively improves the boundary condition, systematically following the asymptotic expansion of the far field. We provide both rigorous error estimates for the method and a range of empirical numerical tests to assess its convergence and robustness.