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Various problems in mechanics lead to nonsmooth (almost) convex minimization problems. Besides contact and friction problems, this structure is exhibited by primal plasticity models and phase-field models of fracture formation. Such problems are typically solved using predictor--corrector or operator-splitting methods. These are expensive, because they solve sequences of linear problems. Also, their convergence behavior is not always clear.

We propose a nonsmooth multigrid method that can solve these problems roughly in the time of one linear problem. For primal plasticity problems this means that solving one spatial problem can be done in the time of a single predictor--corrector iteration. This is shown experimentally, and we prove that the method converges for any initial iterate and any associative smooth or nonsmooth yield law.

The energies used in phase-field models of fracture formation are frequently biconvex rather than convex. Nevertheless, numerical experiments show clear superiority of the multigrid method over traditional operator-splitting schemes. Additionally, a slightly generalized convergence result shows global convergence of the multigrid solver to a stationary point of the energy.