  One of the state-of-the-art strategies for predicting crack propagation,
nucleation, and interaction is the phase-field approach. Despite its
reliability and robustness, the phase-field approach suffers from burdensome
computational cost, caused by the non-convexity of the underlying energy
functional and a large number of unknowns required to resolve the damage
gradients. In this work, we propose to solve such nonlinear systems in a
monolithic manner using the Schwarz preconditioned inexact Newton's (SPIN)
method. The proposed SPIN method leverages the field split approach and
minimizes the energy functional separately with respect to displacement and the
phase-field, in an additive and multiplicative manner. In contrast to the
standard alternate minimization, the result of this decoupled minimization
process is used to construct a preconditioner for a coupled linear system,
arising at each Newton's iteration. The overall performance and the convergence
properties of the proposed additive and multiplicative SPIN methods are
investigated by means of several numerical examples. Comparison with
widely-used alternate minimization is also performed and we show a reduction in
the execution time up to a factor of 50. Moreover, we also demonstrate that
this reduction grows even further with increasing problem size and larger
loading increments.
