In many industrial applications or engineering  problems, contact
between deformable elastic bodies  plays a crucial
  role. As examples we mention incremental forming processes, the 
simulation
of rolling wheels, braking pads on tires and  roller bearings.
  Although early  theoretical results go back to Hertz, there are still
many open problems, and the  numerical simulation of dynamic contact 
problems
remains challenging.
One of the main challenges relates to the fact that the actual contact zone
is  not known a  priori and has to be identified by use of an  iterative 
solver.
Moreover the transition between contact and non-contact is characterized 
by a
change in the type of the boundary condition and thus possibly results in
a solution of reduced regularity.
 From the mathematical point of view, contact problems can be formulated as
free boundary value problems  and analyzed within the abstract framework of
variational inequalities.

This work is an overview of theoretical and numerical results obtained
in recent  years. In particular, we address a priori estimates, iterative
solvers, time integration and a posteriori error bounds. Firstly,
the governing equations and corresponding inequality constraints
for frictional contact problems are stated. Different equivalent 
formulations
are discussed and a weak saddle-point formulation is presented.
Special emphasis is placed on uniformly inf-sup stable finite element
pairings and a suitable approximation of the dual cone.
A priori estimates for the discretization error in the displacement and
in the surface traction are given for linear and quadratic finite 
elements.  Here, we
restrict ourselves to very simple contact settings with given friction 
bounds and
do not take non-matching contact zones into account.
We survey  different solver techniques for the non-linear inequality 
system.
Of special interest are so-called semi-smooth
Newton techniques  applied to an equivalent non-linear system of equations.
  We discuss in detail the structure of the systems to be solved after
consistent linearization and point out the equivalence to classical radial
return mappings. In particular, in the case of no friction the Newton
solver can easily be implemented as a standard primal-dual active set
strategy updating in each iteration step the type and the value of the 
boundary
  condition node-wise.
The next part  is devoted to different
  aspects of adaptive refinement. Bearing in mind that
  the mechanical role of the discrete Lagrange multiplier is that of a
  surface traction, different error indicators can easily be designed. 
However
  the analysis is quite challenging and only a few theoretical results 
exist
taking possibly non-matching meshes and the inequality character
of the formulation into account. Here, we provide upper and lower bounds
for a simplified setting and comment on possible generalizations.
Finally different aspects of time integration are discussed.
For many applications structure-preserving time integration schemes are of
special interest. In this context, energy preservation
is of crucial importance. Unfortunately most of the standard techniques
  result either in very high oscillations in the
dual variable or in  numerical dissipation. We apply a newly combined
time and space integration scheme which is motivated by a reduction of the
index of the differential algebraic system.