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We propose the Truncated Nonsmooth Newton Multigrid Method (TNNMG) as asolver for the spatial problems of the small-strain brittle-fracturephase-field equations. TNNMG is a nonsmooth multigrid method that can solvebiconvex, block-separably nonsmooth minimization problems with linear timecomplexity. It exploits the variational structure inherent in the problem, andhandles the pointwise irreversibility constraint on the damage variabledirectly, 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 severalestablished models of phase-field brittle fracture. We then prove convergenceof the solver to a solution of the nonsmooth Euler--Lagrange equations of thespatial problem for any load and initial iterate. On the way, we show severalcrucial convexity and regularity properties of the models considered here.Numerical comparisons to an operator-splitting algorithm show a considerablespeed increase, without loss of robustness.

Truncated Nonsmooth Newton Multigrid for phase-field brittle-fracture problems, with analysis