  This paper is concerned with a priori error estimates for the local
incremental minimization scheme, which is an implicit time discretization
method for the approximation of rate-independent systems with non-convex
energies. We first show by means of a counterexample that one cannot expect
global convergence of the scheme without any further assumptions on the energy.
For the class of uniformly convex energies, we derive error estimates of
optimal order, provided that the Lipschitz constant of the load is sufficiently
small. Afterwards, we extend this result to the case of an energy, which is
only locally uniformly convex in a neighborhood of a given solution trajectory.
For the latter case, the local incremental minimization scheme turns out to be
superior compared to its global counterpart, as a numerical example
demonstrates.
