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The variational approach to fracture is effective for simulating thenucleation and propagation of complex crack patterns, but is computationallydemanding. The model is a strongly nonlinear non-convex variational inequalitythat demands the resolution of small length scales. The current standardalgorithm for its solution, alternate minimization, is robust but convergesslowly and demands the solution of large, ill-conditioned linear subproblems.In this paper, we propose several advances in the numerical solution of thismodel that improve its computational efficiency. We reformulate alternateminimization as a nonlinear Gauss-Seidel iteration and employ over-relaxationto accelerate its convergence; we compose this accelerated alternateminimization with Newton's method, to further reduce the time to solution; andwe formulate efficient preconditioners for the solution of the linearsubproblems arising in both alternate minimization and in Newton's method. Weinvestigate the improvements in efficiency on several examples from theliterature; the new solver is 5--6$\times$ faster on a majority of the testcases

Linear and nonlinear solvers for variational phase-field models of brittle fracture