In this paper, we consider the numerical solution of a model problem in the form of a parabolic hemivariational inequality that arises in applications of semipermeable media. The model problem is first studied as a particular case of an abstract parabolic hemivariational inequality. A general fully discrete numerical method is introduced for the abstract parabolic hemivariational inequality, where the time derivative of the unknown solution is approximated by the backward divided difference. A Céa’s type inequality is shown as a preparation for error estimation. Then the general result is specialized for the numerical solution of the model problem and an optimal order error estimate with the use of linear finite elements is derived. Finally numerical examples are presented to show the performance of the numerical solutions and the emphasis is to illustrate numerical convergence orders that match the theoretically predicted optimal first order convergence of the linear element solutions with respect to the finite element mesh-size and the time step-size.

在本文中，我们以抛物线半变分不等式的形式考虑模型问题的数值解决方案，该问题在半渗透性介质的应用中出现。首先将模型问题作为抽象抛物线半变分不等式的特殊情况进行研究。针对抽象的抛物线半变分不等式引入了一种通用的全离散数值方法，该方法的未知解的时间导数由向后划分的差近似。显示了Céa类型的不等式，作为误差估计的准备。然后将一般结果专门用于模型问题的数值解，并得出使用线性有限元的最优阶误差估计。