  Numerical simulations of crystal defects are necessarily restricted to finite
computational domains, supplying artificial boundary conditions that emulate
the effect of embedding the defect in an effectively infinite crystalline
environment. This work develops a rigorous framework within which the accuracy
of different types of boundary conditions can be precisely assessed.
  We formulate the equilibration of crystal defects as variational problems in
a discrete energy space and establish qualitatively sharp regularity estimates
for minimisers. Using this foundation we then present rigorous error estimates
for (i) a truncation method (Dirichlet boundary conditions), (ii) periodic
boundary conditions, (iii) boundary conditions from linear elasticity, and (iv)
boundary conditions from nonlinear elasticity. Numerical results confirm the
sharpness of the analysis.
