  This paper is devoted to studying a type of contact problems modeled by
hemivariational inequalities with small periodic coefficients appearing in
PDEs, and the PDEs we considered are linear, second order and uniformly
elliptic. Under the assumptions, it is proved that the original problem can be
homogenized, and the solution weakly converges. We derive an
$O(\epsilon^{1/2})$ estimation which is pivotal in building the computational
framework. We also show that Robin problems--- a special case of contact
problems, it leads to an $O(\epsilon)$ estimation in $L^2$ norm. Our
computational framework is based on finite element methods, and the numerical
analysis is given, together with experiments to convince the estimation.
