  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.
