Motivated by an obstacle problem for a membrane
subject to cohesion forces, constrained minimization problems involving
a non-convex and non-differentiable objective functional
representing the total potential energy are considered. The associated
first order optimality system leads to a hemi-variational inequality,
which can also be interpreted as a special complementarity
problem in function space. Besides an analytical investigation
of first-order optimality, a primal-dual active set solver is introduced.
It is associated to a limit case of a semi-smooth Newton
method for a regularized version of the underlying problem class.
For the numerical algorithms studied in this paper, global as well as
local convergence properties are derived and verified numerically.
The aim of this paper is to devise an adaptive timestep control in the contact--stabilized Newmark method (CONTACX) for dynamical contact problems between two viscoelastic bodies in the framework of Signorini's condition. In order to construct a comparative scheme of higher order accuracy, we extend extrapolation techniques. This approach demands a subtle theoretical investigation of an asymptotic error expansion of the contact--stabilized Newmark scheme. On the basis of theoretical insight and numerical observations, we suggest an error estimator and a timestep selection which also cover the presence of contact. Finally, we give a numerical example.