  The simulation of crack initiation and propagation in an elastic material is
difficult, as crack paths with complex topologies have to be resolved.
Phase-field approach allows to simulate crack behavior by circumventing the
need to explicitly model crack paths. However, the underlying mathematical
model gives rise to a non-convex constrained minimization problem. In this
work, we propose a recursive multilevel trust region (RMTR) method to
efficiently solve such a minimization problem. The RMTR method combines the
global convergence property of the trust region method and the optimality of
the multilevel method. The solution process is accelerated by employing level
dependent objective functions, minimization of which provides correction to the
original/fine-level problem. In the context of the phase-field fracture
approach, it is challenging to design efficient level dependent objective
functions as the underlying mathematical model relies on the mesh dependent
parameters. We introduce level dependent objective functions that combine fine
level description of the crack path with the coarse level discretization. The
overall performance and the convergence properties of the proposed RMTR method
are investigated by means of several numerical examples in three dimensions.
