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.

翻译：我们提出截断非光滑牛顿多重网格法（TNNMG）作为小应变脆性断裂相场方程空间问题的求解器。TNNMG是一种非光滑多重网格方法，能够以线性时间复杂度求解双凸、块可分非光滑极小化问题。该方法利用问题固有的变分结构，直接处理损伤变量上的逐点不可逆性约束，无需正则化或引入局部历史场。本文介绍了该方法，并展示了如何将其应用于相场脆性断裂的几个已有模型。随后，我们证明了对于任意载荷和初始迭代，该求解器均收敛至空间问题的非光滑欧拉-拉格朗日方程的解。在此过程中，我们展示了所考虑模型的若干关键凸性和正则性性质。与算子分裂算法的数值比较表明，该方法在保持鲁棒性的同时显著提升了计算速度。