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.

中文翻译：

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速率无关演化的局部增量最小化方案的先验误差分析

本文涉及局部增量最小化方案的先验误差估计，这是一种用于逼近具有非凸能量的速率无关系统的隐式时间离散化方法。我们首先通过一个反例表明，如果没有对能量的任何进一步假设，就不能期望该方案的全局收敛。对于均匀凸能量类，我们推导出最佳阶次的误差估计，前提是负载的 Lipschitz 常数足够小。之后，我们将此结果扩展到能量的情况，该能量仅在给定解轨迹的邻域中局部均匀凸出。对于后一种情况，如数值示例所示，局部增量最小化方案比其全局对应方案更优越。