This paper is presented for the convergence analysis of the interpolating element-free Galerkin method for the evolutionary variational inequality of the second-order in time, which arises from the theory of viscoelastic materials with edge friction. First, the existence and uniqueness of the solutions for the evolutionary variational inequality of the second-order in time are proved, which are mainly based on the fixed point theorem. Second, the convergence analysis of the interpolating element-free Galerkin method is presented for them. The error estimates show that the convergence order depends not only on the number of basis functions in the interpolating moving least-squares approximation but also the relationship with the time step and the spatial step. Numerical examples verify the convergence analysis and the error estimates.

中文翻译：

---

二阶时间演化变分不等式的无插值Galerkin方法的收敛性分析和误差估计

本文针对二阶时间演化演化不等式的无插值Galerkin方法的收敛性分析进行了研究，该方法是由具有边摩擦的粘弹性材料理论引起的。首先，证明了二阶时间演化变分不等式解的存在性和唯一性，它们主要基于不动点定理。其次，针对它们进行了无插值Galerkin方法的收敛性分析。误差估计表明，收敛阶不仅取决于内插移动最小二乘近似中基函数的数量，而且还取决于与时间步长和空间步长的关系。数值例子验证了收敛性分析和误差估计。