In this article, we addressed the numerical solution of a non-linear evolutionary variational inequality, which is encountered in the investigation of quasi-static contact problems. Our study encompasses both the semi-discrete and fully-discrete schemes, where we employ the backward Euler method for time discretization and utilize the lowest order Crouzeix-Raviart nonconforming finite element method for spatial discretization. By assuming appropriate regularity conditions on the solution, we establish \emph{a priori} error analysis for these schemes, achieving the optimal convergence order for linear elements. To illustrate the numerical convergence rates, we provide numerical results on a two-dimensional test problem.

翻译：本文研究了准静态接触问题中遇到的一类非线性演化变分不等式的数值解法。我们的研究涵盖半离散和全离散格式，其中时间离散采用向后欧拉方法，空间离散采用最低阶Crouzeix-Raviart非协调有限元方法。通过假设解具有适当的正则性条件，我们建立了这些格式的先验误差分析，并在线性单元上达到了最优收敛阶。为展示数值收敛速率，我们给出了二维测试问题的数值结果。