  The "flexible boundary condition" method, introduced by Sinclair and
coworkers in the 1970s, remains among the most popular methods for simulating
isolated two-dimensional crystalline defects, embedded in an effectively
infinite atomistic domain. In essence, the method can be characterized as a
domain decomposition method which iterates between a local anharmonic and a
global harmonic problem, where the latter is solved by means of the lattice
Green function of the ideal crystal. This local/global splitting gives rise to
tremendously improved convergence rates over related alternating Schwarz
methods. In a previous publication (Hodapp et al., 2019, Comput. Methods in
Appl. Mech. Eng. 348), we have shown that this method also applies to
large-scale three-dimensional problems, possibly involving hundreds of
thousands of atoms, using fast summation techniques exploiting the low-rank
nature of the asymptotic lattice Green function.
  Here, we generalize the Sinclair method to bounded domains and develop an
implementation using a discrete boundary element method to correct the infinite
solution with respect to a prescribed far-field condition, thus preserving the
advantage of the original method of not requiring a global spatial
discretization. Moreover, we present a detailed convergence analysis and show
for a one-dimensional problem that the method is unconditionally stable under
physically motivated assumptions. To further improve the convergence behavior,
we develop an acceleration technique based on a relaxation of the transmission
conditions between the two subproblems. Numerical examples for linear and
nonlinear problems are presented to validate the proposed methodology.
