  This paper presents a general convergence theory of penalty based numerical
methods for elliptic constrained inequality problems, including variational
inequalities, hemivariational inequalities, and variational-hemivariational
inequalities. The constraint is relaxed by a penalty formulation and is
re-stored as the penalty parameter tends to zero. The main theoretical result
of the paper is the convergence of the penalty based numerical solutions to the
solution of the constrained inequality problem as the mesh-size and the penalty
parameter approach zero simultaneously but independently. The convergence of
the penalty based numerical methods is first established for a general elliptic
variational-hemivariational inequality with constraints, and then for
hemivariational inequalities and variational inequalities as special cases.
Applications to problems in contact mechanics are described.
