Non-conforming FEM for the quasi-static contact problem

In this article, we addressed the numerical solution of a non-linearevolutionary variational inequality, which is encountered in the investigationof quasi-static contact problems. Our study encompasses both the semi-discreteand fully-discrete schemes, where we employ the backward Euler method for timediscretization and utilize the lowest order Crouzeix-Raviart nonconformingfinite element method for spatial discretization. By assuming appropriateregularity conditions on the solution, we establish \emph{a priori} erroranalysis for these schemes, achieving the optimal convergence order for linearelements. To illustrate the numerical convergence rates, we provide numericalresults on a two-dimensional test problem.