历史相关变分-半变分不等式的数值分析
Numerical Analysis of History-dependent Variational-hemivariational Inequalities
摘要：
本文对接触问题中一类与历史相关的变分-半变分不等式进行了数值分析。提出了三种不同的时间离散化的数值处理方法来近似连续模型。采用不动点迭代算法实现隐格式，并证明了其收敛性，其收敛速度与时间步长和网格尺寸无关。一个特殊的时间离散的历史相关的运营商，导致的数值计划的时间离散系统的唯一的可解性和误差界可以证明没有任何限制的时间步长。在空间逼近方面，采用有限元法，在适当的正则性假设下，给出了线性元解的最优阶误差估计。数值例子来说明理论结果。
英文摘要：
In this paper, numerical analysis is carried out for a class of history-dependent variational-hemivariational inequalities arising in contact problems. Three different numerical treatments for temporal discretization are proposed to approximate the continuous model. Fixed-point iteration algorithms are employed to implement the implicit scheme and the convergence is proved with a convergence rate independent of the time step-size and mesh grid-size. A special temporal discretization is introduced for the history-dependent operator, leading to numerical schemes for which the unique solvability and error bounds for the temporally discrete systems can be proved without any restriction on the time step-size. As for spatial approximation, the finite element method is applied and an optimal order error estimate for the linear element solutions is provided under appropriate regularity assumptions. Numerical examples are presented to illustrate the theoretical results.
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