  In this work, we consider fracture propagation in nearly incompressible and
(fully) incompressible materials using a phase-field formulation. We use a
mixed form of the elasticity equation to overcome volume locking effects and
develop a robust, nonlinear and linear solver scheme and preconditioner for the
resulting system. The coupled variational inequality system, which is solved
monolithically, consists of three unknowns: displacements, pressure, and
phase-field. Nonlinearities due to coupling, constitutive laws, and crack
irreversibility are solved using a combined Newton algorithm for the
nonlinearities in the partial differential equation and employing a primal-dual
active set strategy for the crack irreverrsibility constraint. The linear
system in each Newton step is solved iteratively with a flexible generalized
minimal residual method (GMRES). The key contribution of this work is the
development of a problem-specific preconditioner that leverages the
saddle-point structure of the displacement and pressure variable. Four
numerical examples in pure solids and pressure-driven fractures are conducted
on uniformly and locally refined meshes to investigate the robustness of the
solver concerning the Poisson ratio as well as the discretization and
regularization parameters.
