A-priori error analysis of local incremental minimization schemes for rate-independent evolutions
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Date
2019-07
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Abstract
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.
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Keywords
rate independent evolutions, incremental minimization schemes, a priori error analysis, implicit time discretization, parameterized solutions, differential solutions