  There is currently an increasing interest in developing efficient solvers for
phase-field modeling of brittle fracture. The governing equations for this
problem originate from a constrained minimization of a non-convex energy
functional, and the most commonly used solver is a staggered solution scheme.
This is known to be robust compared to the monolithic Newton method, however,
the staggered scheme often requires many iterations to converge when cracks are
evolving. The focus of our work is to accelerate the solver through a scheme
that sequentially applies Anderson acceleration and over-relaxation, switching
back and forth depending on the residual evolution, and thereby ensuring a
decreasing tendency. The resulting scheme takes advantage of the complementary
strengths of Anderson acceleration and over-relaxation to make a robust and
accelerating method for this problem. The new method is applied as a
post-processing technique to the increments of the solver, hence, the
implementation can be done with minor modifications to already available
software. Moreover, the cost of combining the two acceleration schemes is
negligible. The robustness and efficiency of the method are demonstrated
through numerical examples.
