  We propose the Truncated Nonsmooth Newton Multigrid Method (TNNMG) as a
solver for the spatial problems of the small-strain brittle-fracture
phase-field equations. TNNMG is a nonsmooth multigrid method that can solve
biconvex, block-separably nonsmooth minimization problems with linear time
complexity. It exploits the variational structure inherent in the problem, and
handles the pointwise irreversibility constraint on the damage variable
directly, without regularization or the introduction of a local history field.
In the paper we introduce the method and show how it can be applied to several
established models of phase-field brittle fracture. We then prove convergence
of the solver to a solution of the nonsmooth Euler--Lagrange equations of the
spatial problem for any load and initial iterate. On the way, we show several
crucial convexity and regularity properties of the models considered here.
Numerical comparisons to an operator-splitting algorithm show a considerable
speed increase, without loss of robustness.
