AbstractThis work is concerned with computing the effective crack energy of periodic and random media which arises in mathematical homogenization results for the Francfort–Marigo model of brittle fracture. A previous solver based on the fast Fourier transform (FFT) led to solution fields with ringing or checkerboard artifacts and was limited in terms of the achievable accuracy. As computing the effective crack energy may be recast as a continuous maximum flow problem, we suggest using the combinatorial continuous maximum flow discretization introduced by Couprie et al. The latter is devoid of artifacts, but lacks an efficient large‐scale solution method. We fill this gap and introduce a novel solver which relies upon the FFT and a doubling of the local degrees of freedom which is resolved by the alternating direction method of multipliers (ADMM). Last but not least we provide an adaptive strategy for choosing the ADMM penalty parameter, further speeding up the solution procedure. We demonstrate the salient features of the proposed approach on problems of industrial scale.