  In this work, we consider adaptive mesh refinement for a monolithic
phase-field description for fractures in brittle materials. Our approach is
based on an a posteriori error estimator for the phase-field variational
inequality realizing the fracture irreversibility constraint. The key goal is
the development of a reliable and efficient residual-type error estimator for
the phase-field fracture model in each time-step. Based on this error
estimator, error indicators for local mesh adaptivity are extracted. The
proposed estimator is based on a technique known for singularly perturbed
equations in combination with estimators for variational inequalities. These
theoretical developments are used to formulate an adaptive mesh refinement
algorithm. For the numerical solution, the fracture irreversibility is imposed
using a Lagrange multiplier. The resulting saddle-point system has three
unknowns: displacements, phase-field, and a Lagrange multiplier for the crack
irreversibility. Several numerical experiments demonstrate our theoretical
findings with the newly developed estimators and the corresponding refinement
strategy.
