ABSTRACT A class of one‐dimensional impact and dynamic contact problems taking into account non‐smooth velocities is studied. The new space‐time finite element approximation of dynamic variational inequalities is suggested. The non‐smooth solution for the impact of an obstacle by an elastic bar and its energy conservation under persistency conditions is proved on partitions of the space‐time domain. The explicit solution of the double impact problem is taken as an analytical benchmark for numerical experiments. For an iterative solution, a primal‐dual active set (PDAS) algorithm stemming from semi‐smooth Newton methods is constructed. Numerical experiments compare the space‐time PDAS method with some other existing methods: a mass redistribution method, Paoli–Schatzman scheme, and a Nitsche‐based approximation, highlighting its advantages.
Kovtunenko et al. (2025) studied this question.