Consistent Finite-Dimensional Approximation of Phase-Field Models of Fracture
Provides theoretical justification for numerical methods used in fracture mechanics and image processing.
The paper proves that finite-dimensional approximations of quasi-static evolutions in phase-field fracture models converge to true evolutions as mesh size goes to zero, establishing consistency of numerical schemes for the first time.
In this paper we focus on the finite-dimensional approximation of quasi-static evolutions of critical points of the phase-field model of brittle fracture. In a space discretized setting, we first discuss an alternating minimization scheme which, together with the usual time-discretization procedure, allows us to construct such finite-dimensional evolutions. Then, passing to the limit as the space discretization becomes finer and finer, we prove that any limit of a sequence of finite-dimensional evolutions is itself a quasi-static evolution of the phase-field model of fracture. In particular, our proof shows for the first time the consistency of numerical schemes related to the study of fracture mechanics and image processing.