# A-priori error analysis of local incremental minimization schemes for rate-independent evolutions

## TL;DR

This paper provides a priori error estimates for local incremental minimization schemes used in approximating rate-independent systems, highlighting conditions for convergence and demonstrating the scheme's advantages in non-globally convex energy scenarios.

## Contribution

The paper establishes optimal order error estimates for local incremental minimization schemes under convexity assumptions, extending results to locally convex energies and demonstrating numerical advantages.

## Key findings

- Global convergence cannot be expected without additional assumptions.
- Optimal error estimates are derived for uniformly convex energies with small load Lipschitz constants.
- Local schemes outperform global ones in non-globally convex energy cases, as shown by numerical examples.

## 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.