A dynamic viscoelastic contact problem with normal compliance

A dynamic contact problem between a viscoelastic body and a deformable obstacle is numerically considered in this work. The contact is modeled by using the well-known normal compliance contact condition. The variational formulation of this problem is written in terms of the velocity field and it leads to a parabolic nonlinear variational equation. An existence and uniqueness result is stated. Fully discrete approximations are then introduced by using the finite element method to approximate the spatial variable, and a hybrid combination of the implicit and explicit Euler schemes to discretize the time derivatives. An a priori error analysis is recalled. Then, an a posteriori error analysis is provided extending some results already obtained in the study of the heat equation, other parabolic equations and the quasistatic case. Upper and lower bounds are proved. Finally, some two-dimensional numerical simulations are presented to demonstrate the accuracy and the behavior of the error estimators.